Einstein's theories of special relativity and general relativity form a core part of today's undergraduate (or master's-level) physics curriculum. This is a supplementary problem book or student's manual, consisting of 150 problems in each of special and general relativity i.e., in total 300 problems. The problems have been collected, developed, tested, and refined by the authors over the past two decades from homework and exams given at KTH Royal Institute of Technology, Stockholm, Sweden, starting in the late 1990s. They are a mixture of short-form and multipart extended problems, with hints provided where appropriate. Complete solutions are elaborated for every problem, in a different section of the book; some solutions include brief discussions on their physical or historical significance. The extensive and fully worked out solutions are the main feature of the book and have been revised several times by the authors. Designed as a companion text to complement a main relativity textbook, it does not assume access to any specific textbook. This is a helpful resource for advanced students, for self-study, as a source of problems for university teaching assistants, or as an inspiration for instructors and examiners constructing problems for their lectures, homework, or exams.
mattias blennow is Associate Professor in Theoretical Astroparticle Physics at KTH Royal Institute of Technology, Stockholm, Sweden. His research is mainly directed toward the physics of neutrinos and dark matter and physics beyond the Standard Model. He is the author of the textbook Mathematical Methods for Physics and Engineering (CRC Press, 2018). He has more than 15 years of experience in teaching and has taught special and general relativity both as a lecturer and as a teaching assistant.
tommy ohlsson is Professor of Theoretical Physics at KTH Royal Institute of Technology, Stockholm, Sweden. He has also been a visiting professor at the University of Iceland, Reykjavik, Iceland, over several years. His main research field is theoretical particle physics, especially neutrino physics and physics beyond the Standard Model. He has written the textbook Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory (Cambridge University Press, 2011). He has more than 25 years of university-level teaching experience in relativity theory and physics in general.
300 PROBLEMS IN SPECIAL AND GENERAL RELATIVITY
With Complete Solutions
MATTIAS BLENNOW
KTH Royal Institute of Technology
TOMMY OHLSSON
KTH Royal Institute of Technology
CAMERRDGE
University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA
First published 2022
A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Blennow, Mattias, 1980- author | Ohlsson, Tommy, 1973- author.
Title: 300 problems in special and general relativity : with complete solutions / M. Blennow & T. Ohlsson.
Other titles: Three hundred problems in special and general relativity Description: New York : Cambridge University Press, [2021]|
Includes bibliographical references and index.
Identifiers: LCCN 2021017342 (print) | LCCN 2021017343 (ebook) |
ISBN 9781316510674 (hardback) | ISBN 9781009017732 (paperback) | ISBN 9781009039345 (epub)
Subjects: LCSH: Special relativity (Physics)-Problems, exercises, etc. |
General relativity (Physics)-Problems, exercises, etc. | LCGFT: Problems and exercises.
Classification: LCC QC173.65 .B44 2021 (print) | LCC QC173.65 (ebook)| DDC 530.11-dc23
LC record available at https://lcen.loc.gov/2021017342
LC ebook record available at https://lcen.loc.gov/2021017343
ISBN 978-1-316-51067-4 Hardback
ISBN 978-1-009-01773-2 Paperback
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
To our wives, Ana and Linda
Contents
Preface page ix
Notation, Concepts, and Conventions in Relativity Theory ..... 1
General Notation ..... 1
Special Relativity ..... 3
General Relativity ..... 6
Conventions ..... 12
1 Special Relativity Theory ..... 14
1.1 Basics ..... 14
1.2 Length Contraction, Time Dilation, and Spacetime Diagrams ..... 15
1.3 Lorentz Transformations and Geometry of Minkowski Space ..... 20
1.4 Relativistic Velocities and Proper Quantities ..... 22
1.5 Relativistic Optics ..... 26
1.6 Relativistic Mechanics ..... 29
1.7 Electromagnetism ..... 36
1.8 Energy-Momentum Tensor ..... 41
1.9 Lagrange's Formalism ..... 42
2 General Relativity Theory ..... 43
2.1 Some Differential Geometry ..... 43
2.2 Christoffel Symbols, Riemann and Ricci Tensors, and Einstein's Equations ..... 49
2.3 Maxwell's Equations and Energy-Momentum Tensor ..... 53
2.4 Killing Vector Fields ..... 55
2.5 Schwarzschild Metric ..... 57
2.6 Metrics, Geodesic Equations, and Proper Quantities ..... 58
2.7 Kruskal-Szekeres Coordinates ..... 66
2.8 Weak Field Approximation and Newtonian Limit ..... 68
2.9 Gravitational Lensing ..... 70
2.10 Frequency Shifts ..... 70
2.11 Gravitational Waves ..... 72
2.12 Cosmology and Friedmann-Lemaître-Robertson-Walker Metric ..... 74
3 Solutions to Problems ..... 77
3.1 Solutions to Problems in Special Relativity Theory ..... 77
3.2 Solutions to Problems in General Relativity Theory ..... 185
Bibliography ..... 350
Index to the Problems and Solutions ..... 352
Preface
This book is a supplementary book in the form of a "problem book" or "student's manual" in special and general relativity consisting of a total of 300 problems ( 150 problems each in special and general relativity) with complete and elaborate solutions. It is intended as a companion text to a main textbook, but does not assume any particular textbook. It may be used for self-study act as a source of problems for classes, or as inspiration for teachers and examiners looking to construct new problems for lectures, homework, or exams.
The problems have been collected over the course of about two decades from homework and exams given at KTH Royal Institute of Technology, Stockholm, Sweden, starting in the late 1990s. The extensive and fully worked-out solutions are the main feature of the book and have been revised several times by the authors.
The book is divided into the following chapters:
"Notation, Concepts, and Conventions in Relativity Theory";
Problems in "Special Relativity Theory";
Problems in "General Relativity Theory"; and
"Solutions to Problems" in both special and general relativity,
where the first, unnumbered chapter introduces and sets the stage for both special and general relativity and is intended to be a brief review. The structure of the book is to first present the problems belonging to each main chapter (i.e., Chapters 1 and 2), which are further split into sections in order to obtain a better overview. The solutions are then presented in Chapter 3 (i.e., they do not follow immediately after the problem formulations). The main purpose of this is to suppress the urge for the reader to look at the solution to a problem before making a proper attempt. Some of the problems and solutions are illustrated by figures.
The target audience of the book is students and teachers of special and/or general relativity courses at the master's level that may benefit from it in the way described
above. It will generally be too advanced for the relativity covered by the typical introductory modern physics courses at the bachelor's level, and most likely not advanced enough for an in-depth study at the PhD level.
Finally, we would like to acknowledge our colleagues Jouko Mickelsson, Håkan Snellman, Edwin Langmann, and Teresia Månsson, who have given important contributions to some of the problem statements included in this book. We would also like to thank our editor, Vince Higgs, at Cambridge University Press for a smooth and constructive process with the publication of this book, Torbjörn Bäck for supporting us in developing this book, and Marcus Pernow for proofreading earlier versions of the problem statements and solutions in special relativity. In addition, the KTH Royal Institute of Technology in Stockholm, Sweden, and the University of Iceland in Reykjavik, Iceland, are acknowledged for their hospitality and financial support.
Notation, Concepts, and Conventions in Relativity Theory
This chapter serves to briefly review the concepts relevant to the problems presented in this book. Its purpose is to remind the reader of the basic concepts as well as to introduce the notations and conventions that will be used. In particular, some notations and conventions will vary throughout the different textbooks available on the subject. Some of the different notations have been deliberately used in a number of problems in order to familiarize the reader with the fact that different notations occur in the literature.
General Notation
The components of a vector VV will be written as V^(mu)V^{\mu} in contravariant form and V_(mu)V_{\mu} in covariant form with the index mu\mu running over all the spacetime coordinates. When referring to 3 -vectors, Latin letters will be used for the spatial indices rather than Greek ones, which we use for spacetime coordinates. In the case when an explicit basis for a given vector space is needed, we will use the partial derivatives del_(mu)\partial_{\mu} to denote such a basis, i.e.,
Similarly, tensor components will be denoted with superscripts for contravariant indices and subscripts for covariant indices. Tensors with nn indices, all down, are called covariant tensors of rank nn and tensors with nn indices, all up, are called contravariant tensors of rank nn. Tensors with indices both up and down are so-called mixed tensors. Thus, a vector is a tensor of rank 1 (one index) and a scalar is a tensor of rank zero (no indices). The Einstein summation convention is used throughout
the book, implying that indices that are repeated are to be summed over the relevant range. For example, in a four-dimensional spacetime, we have
where UU and VV are vectors. Contravariant and covariant components are related by lowering and raising with the metric tensor g=(g_(mu nu))g=\left(g_{\mu \nu}\right) and its inverse g^(-1)=(g^(mu nu))g^{-1}=\left(g^{\mu \nu}\right), respectively, i.e.,
{:(0.3)V_(mu)=g_(mu nu)V^(nu)quad" and "quadV^(mu)=g^(mu nu)V_(v):}\begin{equation*}
V_{\mu}=g_{\mu \nu} V^{\nu} \quad \text { and } \quad V^{\mu}=g^{\mu \nu} V_{v} \tag{0.3}
\end{equation*}
Partial derivatives of a given function ff may be denoted in several ways, e.g.,
Several indices after the comma in the latter notation represent higher-order derivatives and the notation may also be used for vector components, for which indices belonging to the vector component are written before the comma and indices denoting derivatives after the comma, i.e.,
{:(0.5)del_(mu)del_(nu)f=f_(,mu nu)quad" and "quaddel_(mu)V_(nu)=V_(nu,mu):}\begin{equation*}
\partial_{\mu} \partial_{\nu} f=f_{, \mu \nu} \quad \text { and } \quad \partial_{\mu} V_{\nu}=V_{\nu, \mu} \tag{0.5}
\end{equation*}
Objects with two indices may be represented in matrix form. We will indicate this by putting parentheses around the considered objects. For example, we can write the object AA with two indices as
By convention, the first index represents the row of the matrix and the second index represents the column. When this is used for one covariant index and one contravariant index, the contravariant index is taken as the row index and the covariant index as the column index.
For objects with more than two indices, we may use matrix notation to represent parts of such objects by inserting a bullet (' ∙\bullet ') in place of the indices being considered. For example, the components A_(1nu)^(mu)A_{1 \nu}^{\mu} of the three-index object A_(lambda nu)^(mu)A_{\lambda \nu}^{\mu} would be represented as the matrix
In a flat 1+31+3-dimensional spacetime and in Cartesian coordinates, the Minkowski metric is given by
{:(0.8)ds^(2)=eta_(mu nu)dx^(mu)dx^(nu)=c^(2)dt^(2)-dx^(2)-dy^(2)-dz^(2):}\begin{equation*}
d s^{2}=\eta_{\mu \nu} d x^{\mu} d x^{\nu}=c^{2} d t^{2}-d x^{2}-d y^{2}-d z^{2} \tag{0.8}
\end{equation*}
where cc is the speed of light in vacuum and x^(0)=ctx^{0}=c t. In units of c=1c=1, so-called natural units, it holds that x^(0)=tx^{0}=t. The metric tensor and its inverse, i.e., the inverse metric tensor, can be written as eta=(eta_(mu nu))=([1,0,0,0],[0,-1,0,0],[0,0,-1,0],[0,0,0,-1])quad<=>quadeta^(-1)=(eta^(mu nu))=([1,0,0,0],[0,-1,0,0],[0,0,-1,0],[0,0,0,-1])\eta=\left(\eta_{\mu \nu}\right)=\left(\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1\end{array}\right) \quad \Leftrightarrow \quad \eta^{-1}=\left(\eta^{\mu \nu}\right)=\left(\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1\end{array}\right).
For any two vectors x=(x^(mu))=(x^(0),x^(1),x^(2),x^(3))x=\left(x^{\mu}\right)=\left(x^{0}, x^{1}, x^{2}, x^{3}\right) and y=(y^(v))=(y^(0),y^(1),y^(2),y^(3))y=\left(y^{v}\right)=\left(y^{0}, y^{1}, y^{2}, y^{3}\right) in Minkowski space described by their contravariant components expressed in Cartesian coordinates, the Minkowski inner product is introduced as
{:(0.10)x*y-=x^(0)y^(0)-x^(1)y^(1)-x^(2)y^(2)-x^(3)y^(3):}\begin{equation*}
x \cdot y \equiv x^{0} y^{0}-x^{1} y^{1}-x^{2} y^{2}-x^{3} y^{3} \tag{0.10}
\end{equation*}
which is obviously commutative, i.e., x*y=y*xx \cdot y=y \cdot x. We also define the notation x^(2)-=x*x=(x^(0))^(2)-(x^(1))^(2)-(x^(2))^(2)-(x^(3))^(2)x^{2} \equiv x \cdot x=\left(x^{0}\right)^{2}-\left(x^{1}\right)^{2}-\left(x^{2}\right)^{2}-\left(x^{3}\right)^{2} for the squared norm ('length') of the vector xx, which is indefinite, since it can be either positive or negative. ^(1){ }^{1} The Minkowski metric eta\eta and its inverse eta^(-1)\eta^{-1} fulfill the relation
where delta_(mu)^(v)\delta_{\mu}^{v} is the Kronecker delta such that delta_(mu)^(v)=1\delta_{\mu}^{v}=1 if mu=v\mu=v and delta_(mu)^(v)=0\delta_{\mu}^{v}=0 if mu!=v\mu \neq v. We can write the Minkowski inner product in multiple ways as
where, e.g., x^(mu)x^{\mu} can be considered as the contravariant components of the vector xx and y_(mu)y_{\mu} the covariant components of the vector yy, i.e., y_(0)=y^(0)y_{0}=y^{0} and y_(i)=-y^(i)y_{i}=-y^{i}, and it also holds that x^(mu)y_(mu)=x_(v)y^(nu)x^{\mu} y_{\mu}=x_{v} y^{\nu}. Furthermore, we say that the vector xx is timelike if x^(2) > 0x^{2}>0, lightlike if x^(2)=0x^{2}=0, and spacelike if x^(2) < 0x^{2}<0. Note that lightlike vectors xx form a cone (x^(0))^(2)=(x^(1))^(2)+(x^(2))^(2)+(x^(3))^(2)\left(x^{0}\right)^{2}=\left(x^{1}\right)^{2}+\left(x^{2}\right)^{2}+\left(x^{3}\right)^{2} and a nonspacelike vector xx is future pointing if x^(0) > 0x^{0}>0 and past pointing if x^(0) < 0x^{0}<0.
In general, a Lorentz transformation Lambda\Lambda between two coordinate systems SS and S^(')S^{\prime} described by coordinates xx and x^(')x^{\prime}, respectively, is given by
{:(0.13)x^(')=Lambda x quad<=>quadx^('mu)=Lambda_(nu)^(mu)x^(nu):}\begin{equation*}
x^{\prime}=\Lambda x \quad \Leftrightarrow \quad x^{\prime \mu}=\Lambda_{\nu}^{\mu} x^{\nu} \tag{0.13}
\end{equation*}
In particular, if the Lorentz transformation is a boost in the x^(1)x^{1}-direction, we can write
where theta\theta is the rapidity, beta-=v//c\beta \equiv v / c, and gamma\gamma is the so-called gamma factor, i.e., gamma-=gamma(v)-=(1-v^(2)//c^(2))^(-1//2)\gamma \equiv \gamma(v) \equiv\left(1-v^{2} / c^{2}\right)^{-1 / 2}, with vv being the relative speed between the coordinate systems SS and S^(')S^{\prime}. Furthermore, it holds that cosh theta=gamma=(1)/(sqrt(1-v^(2)//c^(2))),quad sinh theta=beta gamma=(v)/(c)(1)/(sqrt(1-v^(2)//c^(2))),quad tanh theta=beta=(v)/(c)\cosh \theta=\gamma=\frac{1}{\sqrt{1-v^{2} / c^{2}}}, \quad \sinh \theta=\beta \gamma=\frac{v}{c} \frac{1}{\sqrt{1-v^{2} / c^{2}}}, \quad \tanh \theta=\beta=\frac{v}{c}.
The formulas for (Lorentz) length contraction and time dilation are given by
In the nonrelativistic limit, i.e., v,v^(')≪cv, v^{\prime} \ll c, the classical formula v^('')≃v+v^(')v^{\prime \prime} \simeq v+v^{\prime} is recovered.
Consider radiation of light in a specific coordinate direction of the coordinate system SS. One should think of the radiation as coming from a fixed source in this coordinate system, where the radiation has frequency vv. For an observer in a
coordinate system S^(')S^{\prime} moving along the same coordinate direction with the relative velocity vv, a frequency v^(')v^{\prime} is observed that is given by the relativistic Doppler formula, i.e.,
If the observer is moving away from the source, there is a redshift in the frequency of light, whereas if the observer is moving toward the source, there is a corresponding blueshift.
For the (binary) reaction A+B longrightarrow a+b+cdotsA+B \longrightarrow a+b+\cdots, where two particles with 4-momenta p_(A)p_{A} and p_(B)p_{B} collide, using conservation of 4-momentum, we have
Note that P^(2)P^{2} is invariant for any P=sum_(k=1)^(N)p_(k)P=\sum_{k=1}^{N} p_{k}, where NN is the number of particles, and actually, for any two 4 -vectors AA and BB, the Minkowski inner product A*B=eta_(mu nu)A^(mu)B^(nu)=A^(mu)B_(mu)A \cdot B=\eta_{\mu \nu} A^{\mu} B^{\nu}=A^{\mu} B_{\mu} is invariant under Lorentz transformations. Especially, A^(2)=A^(mu)A_(mu)A^{2}=A^{\mu} A_{\mu} is invariant. This is useful in many applications.
In electromagnetism, the electromagnetic field strength tensor FF is defined as
where A=(A^(mu))=(phi,cA)A=\left(A^{\mu}\right)=(\phi, c \boldsymbol{A}) is the 4 -vector potential with phi\phi and A=A(x)\boldsymbol{A}=\boldsymbol{A}(x) being the electric scalar potential and the magnetic 3 -vector potential, respectively, and can be written as
which is a real antisymmetric matrix, i.e., F^(mu nu)=-F^(nu mu)F^{\mu \nu}=-F^{\nu \mu}, that combines both the electric and magnetic field strengths, i.e., E=(E^(1),E^(2),E^(3))\boldsymbol{E}=\left(E^{1}, E^{2}, E^{3}\right) and B=(B^(1),B^(2),B^(3))\boldsymbol{B}=\left(B^{1}, B^{2}, B^{3}\right). Using this tensor, Maxwell's equations can be written as
where j=(j^(mu))=(rho,j)j=\left(j^{\mu}\right)=(\rho, \boldsymbol{j}) is the 4-current density with rho=rho(x)\rho=\rho(x) and j=j(x)\boldsymbol{j}=\boldsymbol{j}(x) being the charge density and the electric 3-current density, respectively. In addition, we have the two Lorentz invariants
where epsilon_(mu nu lambda omega)\epsilon_{\mu \nu \lambda \omega} is the Levi-Civita tensor with epsilon^(0123)=-epsilon_(0123)=1\epsilon^{0123}=-\epsilon_{0123}=1. Maxwell's equations describe how sources (charges and currents) give rise to electric and magnetic fields. Assuming a moving test charge qq with rest mass mm and parametrizing the trajectory of the test charge as x=x(s)x=x(s), where ss is the proper time parameter, the Lorentz force law describes how the field strengths determine the trajectory of the test charge and is given by
{:(0.28)mc^(2)x^(¨)^(mu)(s)=qx^(˙)_(v)(s)F^(mu v)(x(s)):}\begin{equation*}
m c^{2} \ddot{x}^{\mu}(s)=q \dot{x}_{v}(s) F^{\mu v}(x(s)) \tag{0.28}
\end{equation*}
which is covariant under Lorentz transformations. The energy-momentum tensor TT of the electromagnetic field is defined as
where epsilon_(0)\epsilon_{0} is the electric constant (or permittivity of free space). It holds that TT is symmetric, i.e., T^(mu nu)=T^(nu mu)T^{\mu \nu}=T^{\nu \mu}, and T_(mu)^(mu)=eta_(mu nu)T^(mu nu)=0T_{\mu}^{\mu}=\eta_{\mu \nu} T^{\mu \nu}=0. Furthermore, using Maxwell's equations, we obtain
where f=(f^(mu))=(j*E//c,rho E+j xx B)f=\left(f^{\mu}\right)=(\boldsymbol{j} \cdot \boldsymbol{E} / c, \rho \boldsymbol{E}+\boldsymbol{j} \times \boldsymbol{B}) is the Lorentz force density generated by the 4 -current jj. Without (external) sources, i.e., when j=0,Tj=0, T is conserved, i.e., del_(mu)T^(mu nu)=0\partial_{\mu} T^{\mu \nu}=0.
General Relativity
In a curved spacetime, the metric is defined as
{:(0.31)ds^(2)=g_(mu nu)dx^(mu)dx^(nu):}\begin{equation*}
d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu} \tag{0.31}
\end{equation*}
where the metric tensor and its inverse, i.e., the inverse metric tensor, are given by
respectively. In particular, it holds that grad_(mu)del_(nu)=Gamma_(mu nu)^(lambda)del_(lambda)\nabla_{\mu} \partial_{\nu}=\Gamma_{\mu \nu}^{\lambda} \partial_{\lambda}, where the coefficients Gamma_(mu nu)^(lambda)\Gamma_{\mu \nu}^{\lambda} are called the Christoffel symbols of the second kind. Given a metric g_(mu nu)=g_(nu mu)g_{\mu \nu}=g_{\nu \mu}, the Christoffel symbols of the Levi-Civita connection can be directly computed from
In addition, it holds that Gamma_(mu nu)^(lambda)=Gamma_(nu mu)^(lambda)\Gamma_{\mu \nu}^{\lambda}=\Gamma_{\nu \mu}^{\lambda}, i.e., the Christoffel symbols are always symmetric with respect to the two lower indices.
The parallel transport equation for a vector A^(lambda)A^{\lambda} is given by
Given three vector fields X,YX, Y, and ZZ, the torsion TT and the curvature RR are defined as
{:[(0.40)T(X","Y)=grad_(X)Y-grad_(Y)X-[X","Y]],[(0.41)R(X","Y)Z=[grad_(X),grad_(Y)]Z-grad_([X,Y])Z]:}\begin{align*}
T(X, Y) & =\nabla_{X} Y-\nabla_{Y} X-[X, Y] \tag{0.40}\\
R(X, Y) Z & =\left[\nabla_{X}, \nabla_{Y}\right] Z-\nabla_{[X, Y]} Z \tag{0.41}
\end{align*}
Both the torsion and curvature are tensors and are therefore linear in all of the arguments X,YX, Y, and ZZ, including when the arguments are multiplied by a scalar function ff, e.g.,
respectively. Note that the Levi-Civita connection is torsion free as Gamma_(mu nu)^(lambda)=Gamma_(nu mu)^(lambda)\Gamma_{\mu \nu}^{\lambda}=\Gamma_{\nu \mu}^{\lambda}. For fixed mu\mu and nu\nu, we can write the Riemann curvature tensor in matrix form as
Note that the Riemann curvature tensor is antisymmetric in mu\mu and nu\nu, i.e., R^(∙)_(∙mu nu)=R^{\bullet}{ }_{\bullet \mu \nu}=-R^(∙)_(*v mu)-R^{\bullet}{ }_{\cdot v \mu}, or in component form, R^(omega)_(lambda mu nu)=-R^(omega)_(lambda nu mu)R^{\omega}{ }_{\lambda \mu \nu}=-R^{\omega}{ }_{\lambda \nu \mu}. If the torsion vanishes, i.e., T=0T=0, then we have the first Bianchi identity, i.e.,
Note that it holds that the Einstein tensor is symmetric, i.e., G_(mu nu)=G_(nu mu)G_{\mu \nu}=G_{\nu \mu}, and conserved, i.e., its covariant divergence vanishes grad_(mu)G^(mu nu)=0\nabla_{\mu} G^{\mu \nu}=0. Under local coordinate transformations y=y(x)y=y(x), we have
Symmetries of a spacetime metric are associated to so-called Killing vector fields. Consider a vector field XX. By definition, XX is a Killing vector field if
for all indices mu\mu and nu\nu. Given a Killing vector field X^(mu)X^{\mu} and a geodesic described by coordinate functions x^(mu)(s)x^{\mu}(s), the quantity
is constant along the geodesic.
The dynamics of spacetime in vacuum are described in the Lagrange formalism using the Einstein-Hilbert action, namely
{:(0.60)S_(EH)=-(M_(Pl)^(2))/(2)int Rsqrt(| bar(g)|)d^(4)x:}\begin{equation*}
\mathscr{S}_{\mathrm{EH}}=-\frac{M_{\mathrm{Pl}}^{2}}{2} \int R \sqrt{|\bar{g}|} d^{4} x \tag{0.60}
\end{equation*}
where M_(Pl)-=c^(2)//sqrt(8pi G)M_{\mathrm{Pl}} \equiv c^{2} / \sqrt{8 \pi G} is the Planck mass, RR is the Ricci scalar, and bar(g)=det(g)\bar{g}=\operatorname{det}(g) is the determinant of the metric tensor. For the case of a spacetime not in vacuum, a matter contribution to the action is necessary
where L\mathcal{L} is the Lagrangian density of the matter contribution.
The Einstein gravitational field equations (or simply Einstein's equations) follow from the Euler-Lagrange equations for the action and are given by
where GG is Newton's gravitational constant and T^(mu nu)T^{\mu \nu} is the energy-momentum tensor (or the stress-energy tensor) that describes the distribution of energy in spacetime. The energy-momentum tensor is generally given by
In vacuum, Einstein's equations reduce to G_(mu nu)=0G_{\mu \nu}=0.
In the Newtonian limit and the weak field approximation, i.e., g_(mu nu)≃eta_(mu nu)+g_{\mu \nu} \simeq \eta_{\mu \nu}+h_(mu nu)h_{\mu \nu}, where h_(mu nu)h_{\mu \nu} is a small perturbation, the solutions to Einstein's equations are given by
where Phi\Phi is the gravitational potential for the matter distribution rho\rho and given by Phi=-GM//r\Phi=-G M / r, that is the solution to the Newtonian equation grad^(2)Phi=4pi G rho\nabla^{2} \Phi=4 \pi G \rho. Furthermore, the geodesic equations of motion become
The spherically symmetric vacuum solution to Einstein's equations is the Schwarzschild solution for which the Schwarzschild metric in spherical coordinates is given by
{:(0.68)ds^(2)=g_(mu nu)dx^(mu)dx^(nu)=(1-(2GM)/(c^(2)r))c^(2)dt^(2)-(1-(2GM)/(c^(2)r))^(-1)dr^(2)-r^(2)dOmega^(2):}\begin{equation*}
d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu}=\left(1-\frac{2 G M}{c^{2} r}\right) c^{2} d t^{2}-\left(1-\frac{2 G M}{c^{2} r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2} \tag{0.68}
\end{equation*}
where dOmega^(2)d \Omega^{2} describes the metric on a sphere, i.e., dOmega^(2)=dtheta^(2)+sin^(2)theta dphi^(2)d \Omega^{2}=d \theta^{2}+\sin ^{2} \theta d \phi^{2}. For large rr, the Schwarzschild metric approaches the Minkowski metric. The particular value r=r_(**)-=2GM//c^(2)r=r_{*} \equiv 2 G M / c^{2} represents the Schwarzschild event horizon (or the Schwarzschild radius) and is a coordinate singularity, i.e., it can be removed by a change of coordinates. Such a coordinate change is given by Kruskal-Szekeres coordinates u,v,thetau, v, \theta, and phi\phi, where theta\theta and phi\phi are the ordinary spherical coordinates on a unit sphere S^(2)\mathbb{S}^{2}, the Kruskal-Szekeres metric is given by
{:(0.69)ds^(2)=(16mu^(2))/(r)e^((2mu-r)//(2mu))dudv-r^(2)dOmega^(2)","quad uv=(2mu-r)e^((r-2mu)//(2mu)) < (2GM)/(c^(2)e):}\begin{equation*}
d s^{2}=\frac{16 \mu^{2}}{r} e^{(2 \mu-r) /(2 \mu)} d u d v-r^{2} d \Omega^{2}, \quad u v=(2 \mu-r) e^{(r-2 \mu) /(2 \mu)}<\frac{2 G M}{c^{2} e} \tag{0.69}
\end{equation*}
where mu-=GM//c^(2)\mu \equiv G M / c^{2}.
For a static spacetime, the metric can be written on the form
{:(0.70)ds^(2)=varphi(x)^(2)dt^(2)-g_(ij)(x)dx^(i)dx^(j):}\begin{equation*}
d s^{2}=\varphi(x)^{2} d t^{2}-g_{i j}(x) d x^{i} d x^{j} \tag{0.70}
\end{equation*}
Given two static observers AA and BB in this spacetime, signals sent from AA to BB with frequencies f_(A)f_{A} and f_(B)f_{B}, respectively, will be redshifted according to
In particular, in the Schwarzschild spacetime, a signal sent from a static observer at rr to an observer at infinity will be gravitationally redshifted according to
where UU is the 4 -velocity of the observer and NN the 4 -frequency of the light signal, which is parallel transported along the worldline of the light signal.
In cosmology, the cosmological principles are encoded into the RobertsonWalker metric, which is given by
{:(0.74)ds^(2)=c^(2)dt^(2)-a(t)^(2)((dr^(2))/(1-kr^(2))+r^(2)dOmega^(2)):}\begin{equation*}
d s^{2}=c^{2} d t^{2}-a(t)^{2}\left(\frac{d r^{2}}{1-k r^{2}}+r^{2} d \Omega^{2}\right) \tag{0.74}
\end{equation*}
where a(t)a(t) is some function of the universal time tt and kk is a constant. By a suitable coordinate transformation r|->lambda rr \mapsto \lambda r, one can always choose lambda\lambda such that kk takes one of the three values k=0,+-1k=0, \pm 1. If k=0k=0, then the spatial part for any fixed tt becomes the Euclidean space R^(3)\mathbb{R}^{3}.
From Einstein's equations, the assumption of the Robertson-Walker metric, and the universe being filled by an ideal fluid, follow the two independent Friedmann equations, namely
where the first equation is derived from the 00-component of Einstein's equations and the second equation is derived from the first one together with the trace of Einstein's equations. Here Lambda\Lambda is the cosmological constant.
Conventions
In this book, the following conventions will mainly be used in the presentation of the problems and their corresponding solutions:
Units: We will mostly use units in which the speed of light in vacuum cc has been set to c=1c=1, these are usually known as natural units. In some problems, we have also set ℏ=1\hbar=1, if relevant. Normally, we do not use units in which Newton's gravitational constant GG has been set to G=1G=1. In some problems, it is useful to use SI units.
Vectors: In a four-dimensional spacetime, we will normally denote a 4-vector AA by its contravariant components as follows
where A^(0)A^{0} is the temporal component and A^(i)(i=1,2,3)A^{i}(i=1,2,3) are the spatial components, which is related to the 4 -vector expressed in its covariant components as follows
where it holds that A_(mu)=g_(mu nu)A^(nu)A_{\mu}=g_{\mu \nu} A^{\nu} with (g_(mu nu))\left(g_{\mu \nu}\right) being the given metric tensor. In some textbooks, the convention that the temporal component of a 4 -vector is written as the last component of the vector is used, i.e., A=(A^(1),A^(2),A^(3),A^(4))A=\left(A^{1}, A^{2}, A^{3}, A^{4}\right), whereas in other textbooks, the convention that the standard components of a 4 -vector are chosen as its covariant components might be used. We will not use these conventions.
Metrics: In four-dimensional spacetimes, we adopt the convention that the signature is +--- and, when relevant, place the temporal direction first and denote it by 0 . Therefore, in standard coordinates on Minkowski space, the metric tensor is (eta_(mu nu))=diag(1,-1,-1,-1)\left(\eta_{\mu \nu}\right)=\operatorname{diag}(1,-1,-1,-1) (see Special Relativity) and its inverse is given by (eta^(mu nu))=diag(1,-1,-1,-1)\left(\eta^{\mu \nu}\right)=\operatorname{diag}(1,-1,-1,-1). Thus, we have A_(mu)=eta_(mu nu)A^(nu)A_{\mu}=\eta_{\mu \nu} A^{\nu}, where A_(0)=eta_(00)A^(0)=A^(0)A_{0}=\eta_{00} A^{0}=A^{0} and A_(i)=eta_(ii)A^(i)=-A^(i)A_{i}=\eta_{i i} A^{i}=-A^{i} (for fixed i=1,2,3i=1,2,3 ). In a general coordinates, the metric is given by ds^(2)=g_(mu nu)dx^(mu)dx^(nu)d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu} (see General Relativity) and the metric tensor components represented in matrix form as g=(g_(mu nu))g=\left(g_{\mu \nu}\right) from which its corresponding inverse components g^(-1)=(g^(mu nu))g^{-1}=\left(g^{\mu \nu}\right) can be computed. It must hold that gg^(-1)=g^(-1)g=1_(4)g g^{-1}=g^{-1} g=\mathbb{1}_{4}, where 1_(4)\mathbb{1}_{4} is the 4xx44 \times 4 identity matrix.
Sign convention of the Levi-Civita pseudotensor: We define epsilon^(0123)=+1\epsilon^{0123}=+1 (see Special Relativity), which means that with our convention for the Minkowski metric, we have epsilon_(0123)=-1\epsilon_{0123}=-1.
Partial derivatives: We will mostly denote covariant and contravariant partial derivatives as
where (g^(mu nu))\left(g^{\mu \nu}\right) is the inverse metric tensor.
Covariant derivatives and Christoffel symbols: For covariant derivatives (see General Relativity), we will mostly use the notation grad_(mu)\nabla_{\mu}, but the notation D_(mu)D_{\mu} will sometimes be used. For Christoffel symbols of the second kind, we will only use the notation Gamma_(mu nu)^(lambda)\Gamma_{\mu \nu}^{\lambda} (see General Relativity).
Sign convention of the Ricci tensor: The Ricci tensor may sometimes be defined as R_(mu nu)=R_(mu nu lambda)^(lambda)R_{\mu \nu}=R_{\mu \nu \lambda}^{\lambda}, which introduces a sign difference to our definition (see General Relativity) due to the antisymmetry of the Riemann curvature tensor as
Sign convention of Einstein's equations: There is a sign convention in the definition of Einstein's equations, i.e., G_(mu nu)=+-8pi GT_(mu nu)//c^(4)G_{\mu \nu}= \pm 8 \pi G T_{\mu \nu} / c^{4}, where we use the positive sign.
In general, it is important to keep in mind that different texts may use different conventions. In particular, the sign discrepancies in different expressions will often be due to differing sign conventions of the metric, the Levi-Civita pseudotensor, the Ricci tensor, and Einstein's equations.
Special Relativity Theory
1.1 Basics
Problem 1.1 a) In Figure 1.1, a spacetime diagram for an observer O\mathcal{O} with an inertial frame is shown along with another observer O^(')\mathcal{O}^{\prime} with another inertial frame. The 4 -vectors vec(A), vec(B), vec(U)\vec{A}, \vec{B}, \vec{U}, and vec(V)\vec{V} are drawn. Which of the following statements are true?
In O\mathcal{O} 's inertial frame, the scalar product between vec(A)\vec{A} and vec(B)\vec{B} is zero.
In O^(')\mathcal{O}^{\prime} 's inertial frame, the scalar product between vec(A)\vec{A} and vec(B)\vec{B} is zero.
The scalar product between vec(A)\vec{A} and vec(B)\vec{B} is always nonzero.
In O\mathcal{O} 's inertial frame, the scalar product between vec(U)\vec{U} and vec(V)\vec{V} is zero.
In O^(')\mathcal{O}^{\prime} 's inertial frame, the scalar product between vec(U)\vec{U} and vec(V)\vec{V} is zero.
The scalar product between vec(U)\vec{U} and vec(V)\vec{V} is always nonzero.
b) Which of the 4 -vectors in a) could be proportional to a 4 -velocity? Explain why.
Problem 1.2 Show that
a) every 4 -vector (i.e., vector in Minkowski space) that is orthogonal to a timelike 4 -vector is spacelike.
b) the sum of two future directed time-like 4 -vectors is another future directed timelike 4-vector.
c) every space-like 4 -vector can be written as the difference between two futuredirected lightlike 4 -vectors.
d) the inner product of two future-directed timelike 4-vectors is positive.
Problem 1.3 In a particular inertial frame, two observers have the 3-velocities v_(1)\boldsymbol{v}_{1} and v_(2)\boldsymbol{v}_{2}, respectively. Find an expression for the gamma factor of observer 2 in the rest frame of observer 1 in terms of these velocities.
Problem 1.4 a) Can a rest frame be chosen for a photon? Explain why!
always
sometimes
never
Figure 1.1 Spacetime diagram for observers O\mathcal{O} and O^(')\mathcal{O}^{\prime}.
b) Can a rest frame be chosen for the center of momentum for a system of two photons? Explain why!
always
sometimes
never
1.2 Length Contraction, Time Dilation, and Spacetime Diagrams
Problem 1.5 a) State, explain, and derive the formula for length contraction in special relativity.
b) State, explain, and derive the formula for time dilation in special relativity.
Problem 1.6 A rod with length of 1 m is inclined 45^(@)45^{\circ} in the xyx y-plane with respect to the xx-axis. An observer with the speed sqrt(2//3)c\sqrt{2 / 3} c approaches the rod in the positive direction along the xx-axis. How long does the observer measure the rod to be and at which angle does (s)he observe it to be inclined relative to its xx-axis?
Problem 1.7 When the primary cosmic rays hit the atmosphere, muons are created at an altitude between 10 km and 20 km . A muon in the laboratory lives on average the time tau_(0)=2.2*10^(-6)s\tau_{0}=2.2 \cdot 10^{-6} \mathrm{~s} before it decays into an electron (or a positron) and two neutrinos. Even though a muon can only move tau_(0)c~~660m\tau_{0} c \approx 660 \mathrm{~m} under the time tau_(0)\tau_{0}, a large fraction of the muons will reach the surface of the Earth. How can this be explained? Make a numerical computation for a muon that moves with velocity 0.999 c0.999 c.
Problem 1.8 An express train passes a station with velocity vv. A measurement of the length of the train can be performed in the following different ways:
a) A "continuum" of linesmen is ordered to align along the track. The two men that see the front or the end of the train pass in front of them when their watches show 12:30 make a mark where they stand. The distance L_(a)L_{a} between the marks is measured.
b) One conductor goes to the front of the train and another goes to the end. When the watches of the conductors show 12:1512: 15, they quickly drive a nail into the track. The linesmen measure the distance L_(b)L_{b} between the nails.
c) The stationmaster inspects the receding train through a pair of binoculars. Through the binoculars the stationmaster sees the front of the train to be at the semaphore AA at the same time as its end is at the railway point BB. The linesmen measure the distance L_(c)L_{c} between AA and BB.
d) The stationmaster uses a radar to measure the length of the train. The arrival times of the radar pulses reflected from the front and end of the receding train are t_(1)t_{1} and t_(2)t_{2}, respectively. The distance L_(d)=(t_(1)-t_(2))c//2L_{d}=\left(t_{1}-t_{2}\right) c / 2 is a measure of the length of the train.
Express L_(a),L_(b),L_(c)L_{a}, L_{b}, L_{c}, and L_(d)L_{d} in terms of L_(0)L_{0}, the rest length of the train.
Problem 1.9 A hitchhiker in the Milky Way sits waiting on a small asteroid when a formidably long express space cruiser passes very close to the asteroid. Just as the rear end is opposite to the hitchhiker, (s)he sees lanterns in the front and in the rear end of the cruiser go on simultaneously. Actually, the rear watchman also saw them go on, but according to his hydrogen maser wristwatch he measured a small time difference of 4*10^(-9)s4 \cdot 10^{-9} \mathrm{~s} between the lightening of the forward and rear lanterns. From the type indication on the cruiser - X2000 - our hitchhiker realized that its length was 2*10^(3)m2 \cdot 10^{3} \mathrm{~m}. Had they known what you know, they could have calculated the speed of the cruiser. What was it, according to Einstein's special theory of relativity?
Problem 1.10 Two lamps, which are separated by the distance ℓ\ell in an inertial coordinate system KK, are switched on simultaneously (in KK ). In another inertial coordinate system K^(')K^{\prime}, an observer measures the distance between the lamps to be ℓ^(')\ell^{\prime} and observes the lamps go on with the time difference tau\tau. Express ℓ\ell in terms of ℓ^(')\ell^{\prime} and tau\tau. Assuming that the inertial coordinate system K^(')K^{\prime} is moving along the axis connecting the two lamps, also find the expression for the relative velocity vv between the two inertial coordinate systems.
Problem 1.11 A rod of length ℓ\ell lies in the xzx z-plane of a coordinate system. If the angle between the rod and the xx-axis is theta\theta, calculate the length of the rod as seen by an observer moving with velocity vv along the xx-axis.
Problem 1.12 Two events AA and BB with coordinates x_(A)x_{A} and x_(B)x_{B} are simultaneous for an observer KK with rest frame SS. Another observer, K^(')K^{\prime}, moving with velocity -u along the xx-axis of SS measures these events to not be simultaneous, but such that BB is earlier than AA by the amount Deltat^(')\Delta t^{\prime}. What is the distance LL between the events AA and BB expressed in the frame of KK if it is L^(')L^{\prime} in the rest frame of KK ?
Problem 1.13 An observer SS with rest frame KK observes two events x_(alpha)x_{\alpha} and x_(beta)x_{\beta}. The alpha\alpha event takes place at the origin and the beta\beta event 2 years later at a distance of 10 light years (ly) forward along the x^(1)x^{1}-axis. Another observer S^(')S^{\prime} with rest frame K^(')K^{\prime} moves with velocity vv along the x^(1)x^{1}-axis of KK, passing SS at the origin. The observer S^(')S^{\prime} instead observes the beta\beta event 1 year later than the alpha\alpha event.
a) How far away does S^(')S^{\prime} find the beta\beta event?
b) What is the relative velocity between SS and S^(')S^{\prime} ?
Problem 1.14 The ratio R(mu//e)R(\mu / e) of muon neutrinos to electron neutrinos measured at ground level from the cosmic radiation is R(mu//e)=2R(\mu / e)=2 at low energies. These neutrinos come from the decay of pions, created by the primary cosmic radiation, which consists mostly of protons. The relevant reaction chain can be written in simplified form as follows
As we can see there are two muon neutrinos v_(mu)v_{\mu} produced for every electron neutrino v_(e)v_{e}. When the energy of the muon neutrinos, and therefore the muons, is high enough this ratio goes up, since the muons hit the Earth before they decay, and no electron neutrinos are produced. In the muon's rest frame, the muon lifetime is tau_(0)=2.2 mus\tau_{0}=2.2 \mu \mathrm{~s}. The speed of light is 3*10^(8)m//s3 \cdot 10^{8} \mathrm{~m} / \mathrm{s}. What is the smallest energy of the muons that hit the ground before they decay substantially if they are produced at an altitude of 10 km above ground? The rest mass of the muon is 106 MeV .
Problem 1.15 A circular accelerator has a radius of 50 m . How many turns can a muon take on average in this ring before it decays if its energy is kept constant at 1 GeV ? The average lifetime of the muon in its rest frame is 2.2 mus2.2 \mu \mathrm{~s} and the muon mass is 106 MeV .
Problem 1.16 Consider a triangle at rest in the inertial system KK with sides of length a=3ℓ,b=4ℓa=3 \ell, b=4 \ell, and c=5ℓc=5 \ell in KK.
a) Compute the lengths of the sides and the area of this triangle as measured in an inertial frame K^(')K^{\prime} moving with constant velocity vv parallel to the aa-side of the triangle.
b) Same as in a), but now the observer K^(')K^{\prime} moves parallel to the cc-side of the triangle.
Problem 1.17 Consider a pole of proper length LL moving along the xx-axis in the negative direction with a constant velocity so that the pole is parallel to the xx-axis (see Figure 1.2). At a fixed time, an observer at rest at the spatial origin sees (the optical effect is referred to) the front of the pole at an angle pi//3\pi / 3 with the xx-axis, a mark on the pole at an angle pi//4\pi / 4, and the end of the pole at an angle pi//6\pi / 6. What is the quotient rr between the distance from the front of the pole to the mark and the full length of the pole?
Problem 1.18 Two spaceships, which are initially at rest in some common rest frame, are connected by a straight tensionless string. At time t=0t=0 in this frame, both spaceships start to accelerate in the same direction, in the direction of the string, such that their separation is constant in the initial rest frame. Both spaceships agree to stop accelerating once a predetermined time t_(0)t_{0} has passed in the initial rest frame.
a) Does the string break, i.e., does the distance between the two spaceships increase in the new rest frame of the spaceships?
b) If the distance between the spaceships is originally 40km,t_(0)=30s40 \mathrm{~km}, t_{0}=30 \mathrm{~s}, and the spaceships have constant acceleration of 1//50c//s1 / 50 \mathrm{c} / \mathrm{s} in the initial rest frame, what is the
Figure 1.2 The pole is moving in the negative direction of the xx-axis with a constant velocity vv. The zz-direction is neglected.
distance between the two spaceships in the frame of the leading spaceship after the engines are turned off?
Problem 1.19 Professor A. Einstein is traveling in a train on a rainy night. He is situated in the exact middle of the train, and suddenly lightning strikes right next to him. The train has reflectors in the rear and front, and since the reflections from rear and front reach him at the same time, he falls into slumber convinced that the reflections happened at the same time and that the speed of light is the same in both directions. What he did not see was that Professor W. Wolf was standing on the ground, also next to the lightning strike, observing the events. Draw spacetime diagrams showing how the light signals travel in each of the professors' rest frames. Use these to answer (including motivation) the following
a) Does Professor Wolf see the light reflections reaching Professor Einstein at the same time?
b) Would Professor Wolf agree with Professor Einstein that the light signals were reflected at the same time?
c) Do the reflections reach Professor Wolf at the same time?
Problem 1.20 Two rockets with rest lengths LL and 2L2 L, respectively, move with constant velocities on an interstellar highway. Since the velocities are different, the rockets will pass each other. Call the event when the front of the faster rocket reaches the slower rocket AA and the event when the end of the faster rocket reaches the front of the slower rocket BB (see Figure 1.3). In each rocket there is an observer. Draw one or more spacetime diagrams describing the events, and use it/them to deduce which observer will consider time between AA and BB to be larger (the observer in the short rocket or the observer in the long rocket).
Problem 1.21 Muons created by cosmic rays hitting the atmosphere have a lifetime of 2.2*10^(-6)s2.2 \cdot 10^{-6} \mathrm{~s}. If the muons are created at a height of 10 km , the time to reach the surface of the Earth (measured in the rest frame of the Earth) is at least 10km//c≃3*10^(-5)s10 \mathrm{~km} / c \simeq 3 \cdot 10^{-5} \mathrm{~s}, yet a large fraction of the muons can be measured at sea level.
Figure 1.3 Two rockets with rest-lengths LL and 2L2 L, respectively. Part (a) of the figure shows the event AA, whereas part (b) shows the event BB.
Figure 1.4 Part (a) of the figure shows the dimensions of a guillotine blade in its rest frame, whereas part (b) shows the event of the Scarlet Pimpernel riding by on his horse at velocity vv ( S^(')S^{\prime} is the rest frame of the Scarlet Pimpernel and his horse) and the guillotine blade falls at velocity uu in its own rest frame.
Explain qualitatively why this occurs by describing the situation using spacetime diagrams.
Problem 1.22 During the French Revolution, guillotines with a slanted blade were used to decapitate nobility. The guillotine blade at rest has the dimensions shown in Figure 1.4. Eager to save the nobility, the Scarlet Pimpernel rides by on his horse at velocity vv. How fast does he have to ride in order for the guillotine blade to be horizontal in his rest frame S^(')S^{\prime} if it falls at velocity uu in the guillotine rest frame?
Problem 1.23 In an inertial frame SS, two lights located on the positive xx-axis are moving in the negative xx-direction at speed vv. An observer placed in the origin of SS looks at the light signals coming from the lights. What is the distance between the seen positions of the lights if their separation in their common rest frame is ℓ_(0)\ell_{0} ?
Note: The problem is asking for the separation as seen by the observer, not the actual distance between the lights at a given time.
1.3 Lorentz Transformations and Geometry of Minkowski Space
Problem 1.24 Verify directly from the form of the Lorentz transformation representing a boost in the xx-direction that any object traveling at speed cc in an inertial frame SS travels at speed cc in the boosted frame.
Problem 1.25 A train passes a station just after sunset. The length of the train is LL. In the front and in the rear, it has two lanterns. The lanterns are turned on simultaneously in the train's rest frame. A stationman observes the train pass with velocity vv. Does the stationman see the lanterns go on simultaneously? If not, what is the time difference between the turning on of the two lanterns for the stationman, expressed in terms of LL and vv ?
Problem 1.26 An observer OO on a train of length LL and velocity vv relative to the ground is standing at a distance xL(0 <= x <= 1)x L(0 \leq x \leq 1) from the front AA of the train. When the light from the lamps at AA and BB, at the rear, reach him/her simultaneously, (s)he can calculate at which times t_(1)(A)t_{1}(A) and t_(2)(B)t_{2}(B) they turned on. Another observer O^(')O^{\prime} on the ground can also determine these two times t_(1)^(')t_{1}^{\prime} and t_(2)^(')t_{2}^{\prime} in his rest frame, where the light reaches him as OO just passes him. If (s)he then finds that t_(1)^(')=t_(2)^(')t_{1}^{\prime}=t_{2}^{\prime}, it turns out that the velocity vv of the train can be expressed as a rather simple function of xx. Find this function and show that if v=0v=0, then x=1//2x=1 / 2.
Problem 1.27 A particle of mass mm and energy EE falls from zenith to the Earth along the zz-axis in the rest frame of observer KK. Another observer, K^(')K^{\prime}, moves with velocity vv along the positive xx-axis of KK and will observe the particle to approach K^(')K^{\prime} with an angle theta\theta relative to the z^(')z^{\prime}-axis.
a) Calculate the angle theta\theta expressed in terms of the velocity uu of the particle and the velocity vv of K^(')K^{\prime}.
b) Based on the result of a) give a description of how the starry sky would look like for a space cruiser moving with high speed in our galaxy.
Problem 1.28 Consider a particle with 4-velocity V=gamma(v^('))(c,v^('),0,0)V=\gamma\left(v^{\prime}\right)\left(c, v^{\prime}, 0,0\right). By making a Lorentz transformation with velocity -v-v along the x^(1)x^{1}-axis, show that you can obtain the formula for relativistic addition of velocities, by expressing the velocity v^('')v^{\prime \prime} of the particle in the new system in terms of the velocity v^(')v^{\prime} in the old system and the velocity vv of the motion of the observer.
Problem 1.29 Consider an equilateral triangle with sides of length ℓ\ell, which is at rest in the inertial coordinate system KK. Assume that one of the sides in the triangle is parallel to the x^(1)x^{1}-axis of KK. In an inertial coordinate system K^(')K^{\prime} moving relative to KK with velocity vv along the positive x^(1)x^{1}-axis of KK, an observer measures the lengths of the sides and angles in the triangle. What expressions in ℓ\ell and vv for the lengths and angles does the observer find?
Problem 1.30 An observer K^(')K^{\prime} is moving with constant speed vv along the positive x^(1)x^{1}-axis of an observer KK. A thin rod is parallel to the x^('1)x^{\prime 1}-axis and moving in the direction of the positive x^('2)x^{\prime 2}-axis with relative velocity uu. Show that according to the observer KK the rod forms an angle phi\phi with the x^(1)x^{1}-axis, with
Problem 1.31 A cylinder is rotating around its axis with angular velocity omega\omega (rad/s) in an inertial system. A straight line is drawn along the length of the cylinder. Show that the observer in an inertial system, which moves with velocity vv parallel to the direction of the cylinder axis, will measure the line as twisted around the cylinder. Determine the twist angle per unit length.
Problem 1.32 A fast train (velocity vv ) is passing a station during the night. As the train passes the station, all compartment lights are turned on simultaneously with respect to the rest frame of the train. Relative to an observer standing at the station, the lights seem to be turned on at various times. Compute the velocity uu of the line separating the illuminated and unilluminated parts of the train in the station rest frame.
Problem 1.33 A planet is moving along a circular orbit (radius RR and angular velocity omega\omega ) around a star. A space ship is passing by the star, orthogonal with respect to the plane of motion of the planet, with velocity vv. Compute the orbit of the planet in the rest frame coordinates of the space ship.
Problem 1.34 An observer BB is moving with constant velocity vv along the positive x^(1)x^{1}-axis in the rest frame KK of an observer AA. An observer CC is moving with constant velocity v^(')v^{\prime} along the positive x^('2)^(2){x^{\prime 2}}^{2}-axis in the rest frame K^(')K^{\prime} of the observer BB. Compute the absolute value of the relative velocity of CC with respect to AA. What is the time interval Delta t\Delta t between two events E_(1)E_{1} and E_(2)E_{2} that occur at the same spatial point with time difference Deltat^('')\Delta t^{\prime \prime} in the rest frame K^('')K^{\prime \prime} of observer CC.
Hint: It is sufficient to compute the time coordinate x^(''0)x^{\prime \prime 0} of CC as a function of the coordinates x^(mu)x^{\mu} of AA.
Problem 1.35 Let xx be a lightlike vector in Minkowski space. Show that
where NN is a real normalization factor, uu is a spinor that satisfies X prop uu^(**)X \propto u u^{*}, where XX is a complex 2xx22 \times 2 matrix, so that det X=det(uu^(**))=0\operatorname{det} X=\operatorname{det}\left(u u^{*}\right)=0. Normalize this spinor by the requirement that tr X=2x^(0)\operatorname{tr} X=2 x^{0}.
A Lorentz transformation along the 3-axis is given by
where L(a(v))xL(a(v)) x is the Lorentz-transformed vector and uu is the normalized spinor.
Problem 1.36 Use Einstein's postulate to derive the expressions for a Lorentz boost in the xx-direction.
Problem 1.37 In an inertial frame SS, rockets AA and BB traveling with velocities vv and -v-v, respectively, pass each other at time t=0t=0 at the spatial origin. A time t_(0)t_{0} later, light signals are sent from the origin toward each of the spaceships. Compute the time difference between the spaceships receiving the light signals in the rest frame of one of the rockets.
1.4 Relativistic Velocities and Proper Quantities
Problem 1.38 a) Explain the concept of "relativity of simultaneity." Illustrate it in a spacetime diagram.
b) The worldline of a massive particle in Minkowski space is described by the following equations in some inertial frame (x^(mu))=(ct,x,y,z)\left(x^{\mu}\right)=(c t, x, y, z),
{:(1.5)x(t)=(3)/(2)at^(2)","quad y(t)=2at^(2)","quad z(t)=0:}\begin{equation*}
x(t)=\frac{3}{2} a t^{2}, \quad y(t)=2 a t^{2}, \quad z(t)=0 \tag{1.5}
\end{equation*}
where aa is constant and 0 <= t <= t_(0)0 \leq t \leq t_{0} for some value of t_(0)t_{0}. Compute the particle's 4 -velocity and 4 -acceleration components. What values of t_(0)t_{0} are possible and why? Compute the proper time along this worldline from t=0t=0 to t=t_(0)t=t_{0}.
Problem 1.39 A rod moves with velocity vv along the positive xx-axis in an inertial frame SS. An observer at rest in SS measures the length of the rod to be LL. Another observer moves with the velocity -v-v along the xx-axis. What length, expressed as a function of LL and vv, will this observer measure for the rod? The measurement is done as usual with the endpoints being measured simultaneously for each observer in their respective frames.
Problem 1.40 The worldline of a particle is described by the coordinates x^(mu)(t)x^{\mu}(t) in the system SS. An observer at rest in the system S^(')S^{\prime}, with velocity uu along the positive x^(2)x^{2}-axis relative to SS, measures the velocity of the particle at time t^(')t^{\prime}. Express his result as a function of the velocity of the particle in SS and uu.
Problem 1.41 A spaceship is moving away from Earth. The effect of the engines is regulated so that the the passengers feel the constant acceleration gg. Calculate the distance between the Earth and the spaceship (measured in the rest frame of the Earth) as a function of
a) the time on Earth.
b) the time on the spaceship.
The commander of the spaceship is 40 years of age at the beginning of the voyage. How old is (s)he when the spaceship reaches the Andromeda Galaxy, which lies about 2500000 light years away from Earth?
Hint: 1 year ~~pi*10^(7)s\approx \pi \cdot 10^{7} \mathrm{~s} and g~~10m//s^(2)g \approx 10 \mathrm{~m} / \mathrm{s}^{2}.
Problem 1.42 A rocket (with rest mass m_(0)m_{0} ) starts from rest at the origin of a coordinate system KK. Its velocity along the positive xx-axis is increased by shooting
matter from the rocket with constant velocity ww relative to the instantaneous rest frame of the rocket in the negative xx-direction. Compute the remaining mass mm of the rocket as a function of its velocity vv with respect to the origin of KK.
Problem 1.43 You and your friend are in intergalactic space (assume the Minkowski metric). You leave simultaneously from a space station, with equal speeds vv, in orthogonal directions. Neglect acceleration. After a time TT has passed in your inertial frame, you want to send a message to your friend using a light signal. In which direction (in your rest frame) should you send it?
Problem 1.44 a) In an inertial frame SS, an object travels with 3-velocity uu. A different inertial frame S^(')S^{\prime} is moving in the negative xx-direction relative to SS with relative speed v^(')v^{\prime}. Write down the 4 -velocities of the object and S^(')S^{\prime} in SS.
b) Show that the gamma factor of an object in any inertial frame is given by the inner product of the 4 -velocity of the object and the 4 -velocity of an object at rest in the frame.
c) Express the gamma factor of the object in a) in the frame S^(')S^{\prime} using the result from b).
Problem 1.45 Show that the 4-velocity V^(mu)=dx^(mu)//d tauV^{\mu}=d x^{\mu} / d \tau and 4-acceleration A^(mu)=A^{\mu}=dV^(mu)//d taud V^{\mu} / d \tau of an object are always perpendicular, where tau\tau is the proper time of the object such that V^(2)=1V^{2}=1.
Problem 1.46 You are traveling in your spaceship in flat intergalactic space such that special relativity can be used. You are on your way to a space station when you suddenly discover an enemy spaceship on your radar. You immediately send out a light signal for help to the space station. When you send out your signal, the distance in your coordinate system to the space station is 1 light day. You have a relative speed of c//4c / 4 toward the space station. When the space station receives your signal, they send out a rescue spaceship with a speed 3c//43 c / 4 relative the space station. How long does it take before you get help (as measured by your own clock)? First, how long does it take your light signal to reach the space station, and second, how long does it take for the rescue spaceship to reach you?
Problem 1.47 On an interstellar highway there is a speed limit of uu relative to a reference frame SS. A member of the intergalactic police force is at rest in this system when a spaceship passes at constant velocity v > uv>u (see Figure 1.5). Eager to do the job properly, the police officer starts the pursuit, accelerating with constant proper acceleration aa. The pursuit ends when the police officer catches up with the criminal.
a) How long does the pursuit take according to the criminal?
b) How long does the pursuit take according to the police officer?
c) What is the relative velocity between the police officer and the criminal at the end of the pursuit?
Problem 1.48 An astronaut on an accelerated spaceship uses a coordinate system ( T,X,Y,ZT, X, Y, Z ) related to an inertial system (t,x,y,z(t, x, y, z ) as follows (we set c=1c=1 )
Figure 1.5 A space ship passing the intergalactic police at a relative constant velocity vv.
a) Compute the metric tensor in the astronaut's coordinate system. (The metric in the inertial system is, of course, ds^(2)=dt^(2)-dx^(2)-dy^(2)-dz^(2)d s^{2}=d t^{2}-d x^{2}-d y^{2}-d z^{2}.)
b) Let (k^(mu))=(omega,omega cos(theta),0,omega sin(theta))\left(k^{\mu}\right)=(\omega, \omega \cos (\theta), 0, \omega \sin (\theta)) be the 4 -wavevector of a photon emitted at time t=t_(0)t=t_{0} at the position x=x_(0),y=z=0x=x_{0}, y=z=0 in the abovementioned inertial system. Compute the components of this 4 -wavevector in the astronaut's coordinate system.
c) Compute the duration of a trip of the spaceship on the astronaut's watch (i.e., the proper time) if the trajectory of his spaceship on this trip is, in the astronaut's coordinate system, X(T)=X_(0)=X(T)=X_{0}= constant, Y(T)=vTY(T)=v T for some constant v > 0v>0, Z(T)=0Z(T)=0, and 0 <= T <= T_(0)0 \leq T \leq T_{0}.
Problem 1.49 A rocket AA is accelerating with constant proper acceleration alpha\alpha such that its worldline is given by t^(2)-x^(2)=-1//alpha^(2)t^{2}-x^{2}=-1 / \alpha^{2} in the reference frame SS. Assume that S^(')S^{\prime} is a different reference frame related to SS by a Lorentz transformation in standard configuration with velocity vv. What is the coordinate acceleration a^(')a^{\prime} in the system S^(')S^{\prime} at time t^(')=0t^{\prime}=0 ?
Problem 1.50 Two observers AA and BB are initially colocated at rest in the inertial system SS. At a given time in SS, observer BB starts accelerating with a proper acceleration alpha\alpha. A time t_(0)t_{0} later (as measured by AA ), a light signal is sent from AA toward BB. Find an expression for the proper timed elapsed for observer BB when BB receives the signal. Discuss the limiting cases.
Problem 1.51 Particles in a circular accelerator are accelerated by an electromagnetic field in such a way that they are kept in a circular orbit with constant velocity. What are the corresponding 4-acceleration and proper acceleration of the particle and what is the eigentime required for the particles to complete one orbit? Introduce any quantities required to solve the problem.
Problem 1.52 An object of internal energy MM moving with 4-velocity VV is being acted upon by a force F=fUF=f U, where the known 4-vector UU fulfills U^(2)=1U^{2}=1 and ff is a scalar. How fast is the internal energy of the object increasing (with respect to its proper time) and what is the proper acceleration of the object?
Problem 1.53 Consider a 4-force F^(mu)=(0,f)^(mu)F^{\mu}=(0, \boldsymbol{f})^{\mu} acting on an object of rest energy mm with 3-velocity v\boldsymbol{v}. Compute the rate of change in the rest energy dm//d taud m / d \tau and the
Figure 1.6 An observer oo moving at velocity vv and a particle pp moving toward the observer oo at velocity uu in a direction that makes an angle theta\theta with the direction of the velocity vv.
proper acceleration alpha\alpha, where tau\tau is the proper time of the object worldline. Discuss the requirements on ff for the 4 -force to be a pure force.
Problem 1.54 Given an object with acceleration a_(0)\boldsymbol{a}_{0} in its instantaneous rest frame SS, find an expression for the acceleration in the inertial frame S^(')S^{\prime}, which is moving in the xx-direction with velocity vv relative to SS. What is the maximal and minimal acceleration in S^(')S^{\prime} depending on the direction of the acceleration based on your expression?
Problem 1.55 A particle has 4-momentum P=(E,p)P=(E, p) in an inertial frame SS. An observer is moving by with velocity vv in the xx-direction. Compute the total energy this observer will measure for the particle and the velocity of the particle in the x^(')x^{\prime}-direction of the observer's rest frame.
Problem 1.56 An object originally at rest with mass m(0)=m_(0)m(0)=m_{0} is affected by a constant 4-force F^(mu)=f(1,1)^(mu)F^{\mu}=f(1, \mathbf{1})^{\mu}. Find the object's mass and the time elapsed in the initial rest frame as a function of the object's proper time tau\tau.
Problem 1.57 An observer oo is moving at velocity vv in the xx-direction in an inertial frame SS. In the same inertial frame, a particle pp hits the observer while traveling at a speed uu in a direction that makes an angle theta\theta with the negative xx direction, see Figure 1.6. What is the speed and angle that the observer will measure for the particle? Verify that your result is consistent with both the ultrarelativistic and nonrelativistic limits, i.e., u rarr1u \rightarrow 1 and u,v≪1u, v \ll 1, respectively.
Problem 1.58 In an inertial frame SS an object starting at rest at t=0t=0 is moving with constant coordinate acceleration aa, i.e., x=at^(2)//2x=a t^{2} / 2. Determine the proper time for the object to reach the speed v_(0)v_{0} in SS and the proper acceleration of the object as a function of the time tt in SS.
1.5 Relativistic Optics
Problem 1.59 In 1851, Fizeau measured the speed of light in running water. His result can be summarized in the formula
{:(1.7)u=u_(0)+kv:}\begin{equation*}
u=u_{0}+k v \tag{1.7}
\end{equation*}
where uu is the speed of light in water, that runs with velocity vv. The speed of light in water at rest is u_(0)u_{0} and the drag coefficient kk is given by
where n=c//u_(0)n=c / u_{0} is the refractive index of water. Explain Fizeau's result!
Problem 1.60 See Problem 1.59. Is Fizeau's result still valid if the water runs perpendicular to the motion of light? If not, what is the correction?
Problem 1.61 In 1965, Maarten Schmidt at the Mount Palomar Observatory could identify the strongly redshift Lyman alpha\alpha line in the spectrum of the quasi-stellar radio source 3C 9. Normally, this line has the wavelength 1215"Å"1215 \AA. Schmidt instead found the value 3600"Å"3600 \AA for this line in this radio source. It is possible to explain the redshift in terms of the Doppler effect. This would imply that 3C 9 moves with an enormous speed relative to our galaxy. Determine a lower bound for the speed of 3C 9 .
Problem 1.62 A plane electromagnetic wave moving along the x^(1)x^{1}-axis has the form
Introduce the angular frequency omega=2pi nu\omega=2 \pi \nu and show that the argument of the wave can be written in the form -x_(mu)k^(mu)-x_{\mu} k^{\mu}, where k=((omega )/(c),(omega )/(c),0,0)k=\left(\frac{\omega}{c}, \frac{\omega}{c}, 0,0\right) is the 4 -wave vector of the light wave traveling along the positive x^(1)x^{1}-axis. Show that this vector is lightlike and deduce the formula for the Doppler shift by calculating the change in angular frequency omega\omega under a Lorentz transformation along the x^(1)x^{1}-axis. What does the formula for the Doppler shift look like expressed in terms of the rapidity theta\theta ?
Problem 1.63 A gamma ray burst (GRB) observed in a cluster of faraway galaxies is time dilated and therefore has a total duration about twice as long as GRBs in nearby galaxies. According to the Hubble law, the recession speed is proportional to the distance to the GRB. Calculate the Doppler redshift z=Delta lambda//lambda_(0)z=\Delta \lambda / \lambda_{0} of a typical spectral line from the distant GRB, where lambda\lambda and lambda_(0)\lambda_{0} are the observed and emitted wavelengths, respectively.
Hint: All GRBs can be considered to have the same duration when measured in their respective rest frames.
Problem 1.64 A person watches two objects with constant velocities on a collision course, i.e., they approach each other on a straight line.
a) Assuming that both objects' velocities have the same absolute value c//2c / 2 in the person's frame of reference, compute the absolute value of the velocity with which a person traveling with the first object sees the other object approaching.
b) Assume that the first object sends a light pulse from a ruby laser, which produces visible light with a wavelength lambda_(0)=694.3nm\lambda_{0}=694.3 \mathrm{~nm}, toward the second object. Compute the wavelength of this light pulse as seen by an observer on the second object.
Problem 1.65 A light source is moving at speed vv and at angle theta\theta relative to the separation between the source and a stationary observer.
a) Consider a light pulse with frequency omega_(0)\omega_{0} in the rest frame of the source and determine the frequency omega\omega measured by the observer.
b) Compute the angle theta\theta for which omega=omega_(0)\omega=\omega_{0}.
Problem 1.66 A large disk rotates at uniform angular speed Omega\Omega in an inertial frame SS. Two observers, O_(1)O_{1} and O_(2)O_{2}, ride on the disk at radial distances r_(1)r_{1} and r_(2)r_{2}, respectively, from the center (not necessarily on the same radial line). They carry clocks, C_(1)C_{1} and C_(2)C_{2}, which they adjust so that the clocks keep time with clocks in SS, i.e., the clocks speed up their natural rates by the Lorentz factors
respectively. By the stationary nature of the situation, C_(2)C_{2} cannot appear to gain or lose relative to C_(1)C_{1}. Deduce that, when O_(2)O_{2} sends a light signal to O_(1)O_{1}, this signal is affected by a Doppler shift omega_(2)//omega_(1)=gamma_(2)//gamma_(1)\omega_{2} / \omega_{1}=\gamma_{2} / \gamma_{1}.
Note that, in particular, there is no relative Doppler shift between any two observers equidistant from the center.
Problem 1.67 A light source is moving with speed vv through an optical medium with refractive index nn. Derive an expression for the ratio between the frequency in the frame of the medium and the frequency in the frame of the source as a function of v,nv, n, and the angle theta\theta between the movement direction of the source and the propagation direction of the light (in the frame of the medium).
Problem 1.68 In an inertial frame SS, a mirror is oriented perpendicular to the xx axis and moving with velocity vv in the xx-direction, see Figure 1.7. A light pulse with frequency omega\omega approaches the mirror at an angle theta_("in ")\theta_{\text {in }} in SS. What is the scattering angle theta_("out ")\theta_{\text {out }} and what frequency does the outgoing light have? Explain what happens when v < -cos theta_("in ")v<-\cos \theta_{\text {in }} ?
Problem 1.69 In an inertial frame SS, a light pulse is being directed at an optical medium with refractive index nn which is moving with velocity vv orthogonal to its surface, see Figure 1.8. In the rest frame of the medium, the light pulse is refracted according to Snell's law. An observer in SS makes the interesting observation that the light pulse is still traveling in the same direction after entering the medium. Compute the index of refraction for the medium in terms of the velocity vv and the angle theta^(')\theta^{\prime} between the initial direction of the light pulse and the direction of motion for SS in the rest frame of the medium.
Figure 1.7 A mirror perpendicular to the xx-axis moving with velocity vv in the xx-direction. The ingoing light has frequency omega\omega and makes an angle theta_("in ")\theta_{\text {in }} with the xx-axis, whereas the outgoing light has frequency omega^(')\omega^{\prime} and makes an angle theta_("out ")\theta_{\text {out }} with the xx-axis.
Figure 1.8 An optical medium with refractive index nn moving with velocity vv orthogonal to its surface. In the inertial frame SS, an observer makes the observation that a light pulse is traveling in the same direction (at an angle theta\theta relative to the velocity vv ) before and after entering the medium.
Problem 1.70 A sine wave propagating in a medium can be described by the function sin(N*x)\sin (N \cdot x) in the medium rest frame. Here, (N^(mu))=(omega,k)\left(N^{\mu}\right)=(\omega, k), where omega\omega is the angular frequency and kk the wave number. Assuming the wave velocity in the medium is uu, the relationship between kk and omega\omega is ku=omegak u=\omega. A source with internal frequency omega_(0)\omega_{0} is moving through the medium with velocity vv. Compute the Doppler shifted frequency omega\omega in the medium rest frame when the waves are traveling in the direction of motion. Also discuss the special cases v=uv=u and v=-uv=-u and make sure that your solution reduces to the classical Doppler formula when v≪1v \ll 1.
Problem 1.71 Using the same setup as in Problem 1.50 and assuming that AA sends the light signal using a frequency omega\omega, compute the frequency observed by BB. In addition, if BB carries a mirror and reflects the signal back at AA, find the frequency observed by AA for the reflected signal.
1.6 Relativistic Mechanics
Problem 1.72 The rest energy of an electron is about 0.51 MeV , i.e., the energy a charged particle, with charge equal to the electron charge, would receive when falling down a potential difference of 0.51 MV . Assuming that the electron is accelerated through a linear accelerator (starting from rest) with a potential difference of 10^(6)V10^{6} \mathrm{~V}. Compute the final velocity of the electron.
Problem 1.73 a) An electron e^(-)e^{-}(with mass m_(e)m_{e} ) collides with a positron e^(+)e^{+}(i.e., the antiparticle of the electron with the same mass m_(e)m_{e} as the electron). Show that they cannot annihilate into a single photon gamma\gamma (a photon has zero mass), i.e., the process e^(-)+e^(+)longrightarrow gammae^{-}+e^{+} \longrightarrow \gamma is impossible due to conservation of energy and momentum.
b) Also show that an electron cannot spontaneously emit a photon.
c) Can an electron colliding with a positron annihilate into two photons? Justify your answer.
Problem 1.74 An elementary particle with mass MM decays into two particles aa and bb with masses m_(a)m_{a} and m_(b)m_{b}, respectively. Calculate the momentum of particle aa in the rest frame of particle bb.
Problem 1.75 A particle AA with mass m_(A)m_{A} decays into two particles BB and CC with masses m_(B)m_{B} and m_(C)m_{C}, respectively. Assume that particle AA has speed v_(A)v_{A} before the decay and that particle BB is at rest after the decay, i.e., p_(B)=0\boldsymbol{p}_{B}=\mathbf{0}. Express the speed v_(A)v_{A} in the masses m_(A),m_(B)m_{A}, m_{B}, and m_(C)m_{C}.
Problem 1.76 Two particles, 1 and 2, with masses m_(1)m_{1} and m_(2)m_{2}, respectively, collide and form a new particle with mass MM. Calculate the mass MM and the velocity v\boldsymbol{v} of this new particle in the rest frame of particle 2 as a function of the velocity v_(1)\boldsymbol{v}_{1} of particle 1 in the rest frame of particle 2 and the masses m_(1)m_{1} and m_(2)m_{2}.
Problem 1.77 a) Two particles with rest masses m_(1)m_{1} and m_(2)m_{2}, respectively, move along the xx-axis in the inertial frame of some observer at uniform velocities u_(1)u_{1} and u_(2)u_{2}, respectively. They collide and form a single particle with rest mass mm moving at uniform velocity uu. Assuming that c=1c=1, prove that
when vv is the velocity of particle 2 as measured in the rest frame of particle 1 .
c) Consider two different situations and in both of the situations the relative velocity vv as defined above is the same, and thus, the rest mass mm is the same in both situations, but in one u_(1)=0u_{1}=0 and in the other m_(1)gamma(u_(1))u_(1)=-m_(2)gamma(u_(2))u_(2)m_{1} \gamma\left(u_{1}\right) u_{1}=-m_{2} \gamma\left(u_{2}\right) u_{2}. What is the difference in total energy for the two situations in the frame of the observer?
Problem 1.78 A pion with mass m_(pi)m_{\pi} and energy E_(pi)E_{\pi} moves along the xx-axis. It decays into a muon with mass m_(mu)m_{\mu} and a neutrino with approximately zero mass. Calculate the energy E_(mu)E_{\mu} of the muon when it moves at a right angle relative to the xx-axis in terms of the velocity of the incoming pion and the masses.
Problem 1.79 A pion with mass m_(pi)m_{\pi} decays into an electron with mass mm and an antineutrino with mass m_(nu)m_{\nu}. Calculate the velocity of the antineutrino in the rest frame of the electron as a function of the masses of the particles and determine the limiting value of this velocity as the mass of the antineutrino goes to zero.
Problem 1.80 In June 1998, the Super-Kamiokande Collaboration in Japan reported that it had found evidence for massive neutrinos. Super-Kamiokande measures so-called atmospheric neutrinos, which are produced in hadronic showers resulting from collisions of cosmic rays with nuclei in the upper atmosphere. Two of the dominating processes in the production of atmospheric neutrinos are
where e^(+)e^{+}is a positron, bar(v)_(mu)\bar{v}_{\mu} is an antimuon neutrino, and v_(e)v_{e} is an electron neutrino.
a) Calculate the kinetic energy of the antimuon, T_(mu^(+))T_{\mu^{+}}, and the absolute value of the 3 -momentum of the muon neutrino, p_(nu_(mu))p_{\nu_{\mu}}, when the pion decays at rest according to the first decay. Despite the small mass of the muon neutrino, neglect it! The mass of the pion is m_(pi)m_{\pi} and the mass of the antimuon is m_(mu)m_{\mu}.
b) How far will one of the antimuons, which are produced in the first decay, travel (on average) in the pion rest frame before it decays according to the second decay? The mean lifetime of an antimuon at rest is tau_(mu)\tau_{\mu}.
Problem 1.81 The pions in the sky that are decaying into muons as in Problem 1.14 are produced in collisions between protons in the primary cosmic rays and nitrogen or oxygen in the air. When a pion with energy of 2 GeV is produced, what energy does the muon have if it continues in the same direction as the pion? The expression can be simplified due to the high energy of the pion. What is the resulting expression? What is the muon energy? The pion has a rest mass of 140 MeV , and the neutrino mass can be neglected.
Problem 1.82 A beam of protons that are accelerated to a very high energy hits a beryllium target and produces a shower of particles. Two detectors are placed in a plane behind the target symmetrically around the proton beam axis. Each detector makes an angle of 45^(@)45^{\circ} with this axis and detects mu^(+)mu^(-)\mu^{+} \mu^{-}-pairs, one type of particle in each detector. When the momentum of each muon is 2.2 GeV , one sees an enhancement in the muon rate. This is interpreted as the production of a resonance RR of mass M_(R)M_{R} that decays into the muons. What is the mass M_(R)M_{R} of this resonance? The muon mass is 106 MeV .
Problem 1.83 A particle with mass MM and 4-momentum p=(E,p)p=(E, \mathbf{p}) moves toward a detector when it suddenly decays and emits a photon in the direction of motion.
Figure 1.9 Scattering of two photons gamma+gamma longrightarrow gamma+gamma\gamma+\gamma \longrightarrow \gamma+\gamma.
The energy registered by the detector is omega\omega. Determine what energy the photon had in the rest frame of the decaying particle.
Problem 1.84 An electron moves with constant velocity toward a positron at rest and they annihilate into two photons. The photons go out with angles phi\phi and -phi-\phi relative to the direction of the incoming electron.
a) Calculate the angle as a function of the total energy of the electron.
b) Show that in the nonrelativistic limit the angle is given by cos phi=v//(2c)\cos \phi=v /(2 c).
Problem 1.85 Two photons with wavelengths lambda_(1)\lambda_{1} and lambda_(2)\lambda_{2}, respectively, are scattered against each other according to Figure 1.9. Calculate the wavelength of the photon with scattering angle theta\theta, i.e., express lambda\lambda as a function of lambda_(1),lambda_(2)\lambda_{1}, \lambda_{2}, and theta\theta.
Hint: p=(h)/( lambda)p=\frac{h}{\lambda}, where hh is Planck's constant.
Problem 1.86 A K-meson with mass MM decays at rest into two charged pions with the same mass mm and a photon according to the reaction formula
The momenta of the particles are given in parentheses after each particle symbol. Calculate the speed vv of the pions in center-of-mass frame where (where p_(1)+p_(2)=0\mathbf{p}_{1}+\mathbf{p}_{2}=0 ) as a function of the masses of the particles and the photon energy k^(0)=omegak^{0}=\omega in the rest frame of the decaying particle.
Problem 1.87 A Sigma^(0)\Sigma^{0} particle with speed c//3c / 3 in the direction toward a gamma detector suddenly decays into a Lambda\Lambda particle and a photon. The photon continues toward the detector.
a) What energy does the Sigma^(0)\Sigma^{0} particle have in the system in which the detector is at rest?
b) What energy does the photon have in the rest system of the Sigma^(0)\Sigma^{0} particle?
c) What energy will be registered in the detector?
The mass of the Lambda\Lambda is m_(Lambda)~~1115.7MeVm_{\Lambda} \approx 1115.7 \mathrm{MeV} and that of Sigma^(0)\Sigma^{0} is m_(Sigma^(0))~~1192.6MeVm_{\Sigma^{0}} \approx 1192.6 \mathrm{MeV}.
Problem 1.88 In elastic scattering of two particles onto each other, the same type of particles are present before and after the collision. Thus, in e+p longrightarrow e+pe+p \longrightarrow e+p elastic
scattering of electrons on protons with corresponding 4-momenta p_(e),p_(p),p_(e)^(')p_{e}, p_{p}, p_{e}^{\prime}, and p_(p)^(')p_{p}^{\prime}, one can form an invariant called tt, defined as t=(p_(e)-p_(e)^('))^(2)t=\left(p_{e}-p_{e}^{\prime}\right)^{2}.
a) Show that, in the center-of-mass system defined by the total 3 -momentum being 0\mathbf{0}, the quantity -t-t equals the square of the change of the 3 -momentum, i.e., -t=(p_(e)-p_(e)^('))^(2)-t=\left(\mathbf{p}_{e}-\mathbf{p}_{e}^{\prime}\right)^{2} and express this quantity in terms of the scattering angle theta\theta between the incoming and outgoing electrons and the modulus of the 3-momentum |p_(e)|\left|\mathbf{p}_{e}\right| of the incoming electron.
b) Calculate the kinetic energy, T_(p)^(')T_{p}^{\prime}, of the outgoing proton in the laboratory system, where the incoming proton is at rest before the collision, in terms of the variable tt.
Problem 1.89 What is the kinetic energy TT of the pion required to create the resonance Delta(1232)\Delta(1232) in the reaction
where pi\pi is a pion and pp is a proton? The proton is at rest before the collision. The result should be expressed in terms of the masses of the particles involved.
Problem 1.90 The scattering probabilities for the reactions pi+d longrightarrow p+p\pi+d \longrightarrow p+p and for the reversed reaction p+p longrightarrow pi+dp+p \longrightarrow \pi+d are related due to so-called time reversal invariance. However, they must be compared at the same center-of-mass energy. Calculate the relation between the kinetic energy T_(pi)T_{\pi} of the pion (pi)(\pi), in the frame where the deuteron (d)(d) is at rest before the collision in the first reaction, and the kinetic energy T_(p)T_{p} of one of the protons ( pp ) in reversed reaction, when the other proton is at rest, respecting the above condition on the center-of-mass energy.
Problem 1.91 Consider the reaction pi^(+)+n longrightarrowK^(+)+Lambda\pi^{+}+n \longrightarrow K^{+}+\Lambda in the rest frame of nn. The masses of the particles are m_(pi^(+)),m_(n),m_(K^(+))m_{\pi^{+}}, m_{n}, m_{K^{+}}, and m_(Lambda)m_{\Lambda}, respectively. What is the kinetic energy TT of the pi^(+)\pi^{+}when the K^(+)K^{+}has total energy EE and moves off at an angle of 90^(@)90^{\circ} to the direction of the incident pi^(+)\pi^{+}? (T:}\left(T\right. should be expressed in m_(pi^(+)),m_(n)m_{\pi^{+}}, m_{n}, m_(K^(+)),m_(Lambda)m_{K^{+}}, m_{\Lambda}, and EE.)
Problem 1.92 The mass of the meson pi^(0)\pi^{0} can be measured by the reaction
where pp is a proton, pi^(-)\pi^{-}is a negative pion, and nn is a neutron. The uncharged pi^(0)\pi^{0} meson decays very quickly into two photons and cannot be easily measured. However, the velocity of the final neutron can be measured and is found to be v_(n)=v_{n}= ( 0.89418+-0.00017)cm//ns0.89418 \pm 0.00017) \mathrm{cm} / \mathrm{ns}. Derive the formula that expresses the mass of the pi^(0)\pi^{0} meson as a function of the masses of the proton, the pi^(-)\pi^{-}, the neutron, and the velocity v_(n)v_{n}, assuming that the reaction takes place at rest for the incoming particles. Simplify the result by showing that the velocity is small, so that we need to retain only lowest nontrivial order in v_(n)//cv_{n} / c.
Problem 1.93 A thermal neutron is absorbed by a proton at rest and a deuteron is formed together with a photon. This exothermic reaction is formally
The binding energy BB of the deuteron is about 2.23 MeV . Calculate, relativistically, the energy of the emitted photon as a function of the masses of the particles and the binding energy BB.
Problem 1.94 A hydrogen atom H , consisting of an electron and a proton with binding energy B=13.6eVB=13.6 \mathrm{eV}, can disintegrate into its two constituent particles by being hit by a photon. The reaction is
Calculate, relativistically, the smallest photon energy in the rest frame of H required for this process to occur expressed in terms of BB and the hydrogen mass m_(H)m_{\mathrm{H}}.
Problem 1.95 Similarly to the cosmic microwave background (CMB) of photons with a temperature of T_(CMB)∼2.7KT_{\mathrm{CMB}} \sim 2.7 \mathrm{~K}, there should be a cosmic neutrino background (CNB) with a temperature of T_(CNB)∼1.9KT_{\mathrm{CNB}} \sim 1.9 \mathrm{~K}. At these temperatures, their kinetic energy is very tiny. Suppose a very high-energy antineutrino would hit such a neutrino and annihilate it. A result of this collision could be the production of a Z^(0)Z^{0} boson which decays hadronically. The reaction is formally
What is the threshold energy for the antineutrino for this to occur? In particular, consider the two limits
a) The CNB neutrinos have a mass of m_(v)=0.15eVm_{v}=0.15 \mathrm{eV}.
b) The CNB neutrinos have very small masses (m_(v)//(k_(B)T)rarr0)\left(m_{v} /\left(k_{B} T\right) \rightarrow 0\right).
Hint: In a gas of particles at temperature TT, the mean kinetic energy of the particles is given by E_(k)=3k_(B)T//2E_{k}=3 k_{B} T / 2, where k_(B)≃8.6*10^(-5)eV//Kk_{B} \simeq 8.6 \cdot 10^{-5} \mathrm{eV} / \mathrm{K} is Boltzmann's constant. The Z^(0)Z^{0} mass is m_(Z^(0))≃91GeVm_{Z^{0}} \simeq 91 \mathrm{GeV}.
Problem 1.96 Consider elastic scattering of photons on electrons
where kk and pp are the incoming photon and electron 4-momenta and k^(')k^{\prime} and p^(')p^{\prime} the corresponding outgoing 4 -momenta.
a) In the laboratory system, the incoming electron is at rest and the outgoing photon is scattered at an angle theta\theta with respect to the direction of the incoming photon. Use invariants to derive the so-called "Compton formula," i.e., the difference between the outgoing and incoming photon wavelengths, as a function of theta\theta, in units c=1c=1 and ℏ=1\hbar=1.
b) Derive the angular frequency (energy) of the outgoing photon in the center-of-mass system in terms of the incoming photon angular frequency (energy) in the laboratory system.
Problem 1.97 In Compton scattering gamma+e longrightarrow e+gamma\gamma+e \longrightarrow e+\gamma, photons of a fixed energy omega\omega are scattered against electrons, which can be considered at rest in the laboratory frame. Compute the kinetic energy of the outgoing electron as a function of the scattering angle theta\theta of the outgoing photon.
Problem 1.98 Inverse Compton scattering occurs when low-energy photons collide with high-energy electrons. Assuming that the photon and electron are originally moving in the same direction, find an expression for the photon energy after the collision as a function of the initial photon energy, the velocity and mass of the electron, and the scattering angle theta\theta of the photon.
Problem 1.99 An antimuon mu^(+)\mu^{+}decays into a positron e^(+)e^{+}and two neutrinos v_(e)v_{e} and bar(v)_(mu)\bar{v}_{\mu}. The reaction is
Give an expression for the largest possible total energy of the electron neutrino nu_(e)\nu_{e} in the rest frame of the antimuon. You may assume that the neutrino masses are negligible compared to lepton masses.
Problem 1.100 A rho\rho-meson with mass m_(rho)≃770MeV//c^(2)m_{\rho} \simeq 770 \mathrm{MeV} / c^{2} sometimes decays into a pair of muons ( mu^(-)\mu^{-}and mu^(+)\mu^{+}) with mass m_(mu^(-))=m_(mu^(+))≃106MeV//c^(2)m_{\mu^{-}}=m_{\mu^{+}} \simeq 106 \mathrm{MeV} / c^{2} and a photon, gamma\gamma. What is the maximal kinetic energy that the mu^(+)\mu^{+}can have in this decay in the rest frame of the rho\rho-meson?
Problem 1.101 There is a possibility that neutrinos are their own antiparticles. If this is true, then the so-called neutrinoless double beta decay
is allowed. Derive expressions for the maximal and minimal possible values of the sum of the kinetic energy of the electrons in the rest frame of ^(76)Ge{ }^{76} \mathrm{Ge}. Express your answer in terms of the particle masses.
Problem 1.102 At the LHC (Large Hadron Collider), two photons are measured with 4-momenta
{:(1.13)p_(1)=omega_(1)(1","1","0","0)quad" and "quadp_(2)=omega_(2)(1","cos theta","sin theta","0):}\begin{equation*}
p_{1}=\omega_{1}(1,1,0,0) \quad \text { and } \quad p_{2}=\omega_{2}(1, \cos \theta, \sin \theta, 0) \tag{1.13}
\end{equation*}
respectively. Assuming that the photon pair results from the decay of a new particle phi\phi such that phi longrightarrow gamma gamma\phi \longrightarrow \gamma \gamma, what is the mass of the new particle?
Problem 1.103 In an accelerator, protons are accelerated until they reach a kinetic energy of 8000 MeV and are then made to collide with protons at rest. If the sum of the kinetic energies of two colliding protons (measured in the center-of-mass system) is larger than the rest energy of a proton-antiproton pair, then such a pair can be formed according to the reaction formula
where pp is a proton and bar(p)\bar{p} is an antiproton.
Is the energy 8000 MeV sufficient for the reaction to go? The rest mass of the proton is 938 MeV .
Problem 1.104 Protons at rest are bombarded with pi\pi-mesons. How large kinetic energy do the mesons need to have for the reaction
to take place? The rest mass of the particles are m_(pi^(-))=m_(pi^(+))~~140MeV,m_(p)~~m_{\pi^{-}}=m_{\pi^{+}} \approx 140 \mathrm{MeV}, m_{p} \approx 938 MeV , and m_(n)~~940MeVm_{n} \approx 940 \mathrm{MeV}.
Problem 1.105 In the CELSIUS ring at the The Svedberg Laboratory in Uppsala, Sweden, one would like to study the reaction
The available kinetic energy of the protons is T_(p)=700MeVT_{p}=700 \mathrm{MeV} and the deuterons (d) can be considered to be at rest. The rest masses of the particles are m_(p)~~m_(n)m_{p} \approx m_{n}, m_(d)~~m_(p)+m_(n),m_(n)=940MeVm_{d} \approx m_{p}+m_{n}, m_{n}=940 \mathrm{MeV}, and m_(eta)=550MeVm_{\eta}=550 \mathrm{MeV}.
a) Is the reaction possible?
b) If the kinetic energy of the protons in the beam is increased to T_(p)=1350MeVT_{p}=1350 \mathrm{MeV}, what is the maximum kinetic energy that the eta\eta can get in the system in which the nucleons are at rest after the reaction, expressed in terms of the rest masses and the kinetic energies?
Problem 1.106 In neutrino detection, the quasi-elastic ( nu_(mu)+X longrightarrow mu+Y\nu_{\mu}+X \longrightarrow \mu+Y, where XX and YY are different nuclei) and 1pi(v_(mu)+X longrightarrow mu+Y+pi^(0))1 \pi\left(v_{\mu}+X \longrightarrow \mu+Y+\pi^{0}\right) processes are relevant at relatively low energies. Compute the ratio between the neutrino threshold energies for these processes in the rest frame of the nucleus XX. Express your answer in terms of the different particle masses (the neutrino may be considered massless for the purposes of this problem).
Problem 1.107 Consider the particle collision e^(-)+e^(-)longrightarrowe^(-)+e^(-)+e^(-)+e^(+)e^{-}+e^{-} \longrightarrow e^{-}+e^{-}+e^{-}+e^{+}. Compute the necessary total energy of one of the initial electrons in the rest frame of the other for this process to occur. Also, compute the ratio between this energy and the total required energy in the center-of-momentum frame.
Problem 1.108 We can produce neutral kaons in a proton collision through the reaction p+p longrightarrow p+p+K^(0)p+p \longrightarrow p+p+K^{0}. Find an expression for the threshold kinetic energy of the protons of this reaction when
a) One proton is stationary in the lab frame (find the threshold kinetic energy of the other proton).
b) Both protons have the same kinetic energy (quote the total kinetic energy of both protons).
Problem 1.109 A particle chi\chi hits a stationary proton pp and undergoes inelastic scattering to a new state chi^(**)\chi^{*} while keeping the proton intact. Determine the threshold kinetic energy of chi\chi for this scattering to occur if m_(chi^(**))=m_(chi)+delta > m_(chi)m_{\chi^{*}}=m_{\chi}+\delta>m_{\chi}. Discuss your result in the limit when delta≪m_(chi)\delta \ll m_{\chi}.
Problem 1.110 Neutrinos are emitted from core collapse supernovae. If a core collapse supernova occurs at a distance LL from Earth and each neutrino has a total energy EE, how much more time would pass (in the rest frame of the Earth) until the neutrinos reach us if they have a small mass m > 0m>0 compared to if they were massless (m=0)(m=0) ? Give an exact answer as well as a reasonable approximation for when m≪Em \ll E.
Problem 1.111 An elementary particle of charge e(ee(e is the elementary charge) is accelerated from rest in a 100 m long straight insulated vacuum cylinder (a linear accelerator) with a constant electric field of 10^(4)V//m10^{4} \mathrm{~V} / \mathrm{m} across the endpoints.
a) What kinetic energy will the particle obtain after the acceleration?
b) How long time does it take for particle to pass through the tube if it starts from rest? Hint: Use the energy as an integration variable.
1.7 Electromagnetism
Problem 1.112 Show by explicit calculation, using the chain rule for derivation and the properties of the Lorentz transformations, that
is invariant under Lorentz transformations, i.e., if A^(mu)(x)A^{\mu}(x) is a solution to Eq. (1.14), then A^('mu)(x^('))A^{\prime \mu}\left(x^{\prime}\right) is a solution to the same equation in the primed variables x^(')=Lambda xx^{\prime}=\Lambda x, where Lambda\Lambda is a Lorentz transformation.
Problem 1.113 Show that the gauge transformation A_(mu)|->A_(mu)^(')=A_(mu)+del_(mu)psiA_{\mu} \mapsto A_{\mu}^{\prime}=A_{\mu}+\partial_{\mu} \psi, where psi\psi is an arbitrary scalar field, does not affect the field tensor F_(mu nu)=del_(mu)A_(nu)-del_(nu)A_(mu)F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}.
Problem 1.114 An inertial coordinate system K^(')K^{\prime} is moving relative to another inertial coordinate system KK with constant velocity vv along the positive x^(1)x^{1}-axis of KK.
a) Assume that a stick of length ℓ\ell is at rest in KK such that Deltax=(ℓ,0,0)\Delta \mathbf{x}=(\ell, 0,0). Calculate Deltax^(')\Delta \mathbf{x}^{\prime} in K^(')K^{\prime}.
b) Assume that there is a constant electric field E=(0,0,E)\mathbf{E}=(0,0, E) in KK (no magnetic field, i.e., B=0\mathbf{B}=\mathbf{0} in KK ). Calculate E^(')\mathbf{E}^{\prime} and B^(')\mathbf{B}^{\prime} in K^(')K^{\prime}.
Problem 1.115 An observer at rest in a frame KK experiences only an electric field E\mathbf{E}. Another observer in another frame K^(')K^{\prime}, moving with velocity vv along the positive xx-axis, will observe a magnetic field B^(')\mathbf{B}^{\prime}. Calculate this magnetic field for small velocities (linear terms in vv ) and show that this field is perpendicular to both the electric field E^(')\mathbf{E}^{\prime} and the velocity of KK relative to K^(')K^{\prime}.
Problem 1.116 Let K,K^(')K, K^{\prime}, and K^('')K^{\prime \prime} be as in Problem 1.34. Assume that there is a constant electric field E=(0,1,0)\mathbf{E}=(0,1,0) (in some given physical units) in the coordinate system KK. We assume that the magnetic field B\mathbf{B} vanishes in KK. Compute the components of both the electric and magnetic fields in the coordinate systems K^(')K^{\prime} and K^('')K^{\prime \prime}.
Problem 1.117 Compute the electric and magnetic field components due to a point charge qq moving with velocity vv along the positive xx-axis.
Problem 1.118 A particle of mass mm and electric charge qq is moving in a constant electric field E\boldsymbol{E}. Use the Lorentz force law to calculate the velocity of the particle as a function of the displacement rr from the origin along the direction of motion. The particle starts off at rest.
Problem 1.119 A current II is flowing through a straight uncharged conductor. Determine the electromagnetic field in an inertial system K^(')K^{\prime} that moves parallel to the conductor with velocity vv
a) by transforming the electromagnetic field tensor from the rest frame KK of the conductor to K^(')K^{\prime},
b) by transforming the current-density 4 -vector from KK to K^(')K^{\prime}, and then, knowing the charge of the conductor and its current relative to KK determine the field in K^(')K^{\prime}.
Problem 1.120 Maxwell's equations can be expressed by means of the electromagnetic 4 -vector potential AA. When del_(mu)A^(mu)=0\partial_{\mu} A^{\mu}=0 (i.e., the Lorenz gauge), they take on a simple form. What is this form? Assuming that Maxwell's equations are in this simple form, and furthermore, J=0J=0 (i.e., current free), show for a plane wave, A^(mu)=epsi^(mu)e^(ik*x)A^{\mu}=\varepsilon^{\mu} e^{i k \cdot x}, where epsi\varepsilon is the polarization vector, that
i.e., the electric and magnetic fields are perpendicular to the direction of motion.
Problem 1.121 Calculate the Lorentz invariants F_(mu nu)F^(mu nu)F_{\mu \nu} F^{\mu \nu} and epsilon_(mu nu omega lambda)F^(mu nu)F^(omega lambda)\epsilon_{\mu \nu \omega \lambda} F^{\mu \nu} F^{\omega \lambda} for a free electromagnetic plane wave A^(mu)(x)=epsilon^(mu)e^(ik*x)A^{\mu}(x)=\epsilon^{\mu} e^{i k \cdot x}, where epsilon\epsilon is the polarization vector. Give a physical interpretation of your result.
Problem 1.122 a) Prove that the scalar product E*B\boldsymbol{E} \cdot \boldsymbol{B} between the electric and magnetic field vectors is invariant under Lorentz transformations.
b) Show that if the electric and magnetic fields E\boldsymbol{E} and B\boldsymbol{B} are orthogonal for one observer, they are orthogonal for any observer.
c) Show that E\boldsymbol{E} and B\boldsymbol{B} are orthogonal for free plane waves with A^(mu)(x)=epsi^(mu)e^(ik*x)A^{\mu}(x)=\varepsilon^{\mu} e^{i k \cdot x}, where epsi\varepsilon is the polarization vector.
d) Show for the plane waves that E xx B=AkE \times B=A k, where kk is the wave vector and AA is a nonvanishing expression.
Problem 1.123 An electron with mass m_(0)m_{0} is moving in a homogeneous magnetic field B=(0,0,B)B=(0,0, B) and no electric field. Calculate its trajectory if it has velocity u=(u,0,0)\boldsymbol{u}=(u, 0,0) at time t=0t=0.
Problem 1.124 In an inertial coordinate system KK, there is a constant electric field E=(cB,0,0)\mathbf{E}=(c B, 0,0) and a constant magnetic field B=(0,B,0)\mathbf{B}=(0, B, 0). In another inertial system K^(')K^{\prime}, the same fields are measured to be E^(')=(0,2cB,cB)\mathbf{E}^{\prime}=(0,2 c B, c B) and the xx-component B_(x)^(')=B_{x}^{\prime}= 0 . Compute B_(y)^(')B_{y}^{\prime} and B_(z)^(')B_{z}^{\prime}.
Problem 1.125 Observer AA measures the electric and magnetic field strengths to be E=(alpha,-alpha,0)\mathbf{E}=(\alpha,-\alpha, 0) and B=(0,0,2alpha//c)\mathbf{B}=(0,0,2 \alpha / c), respectively, where alpha!=0\alpha \neq 0. Another observer, observer BB, makes the same measurements and finds E^(')=(0,0,2alpha)\mathbf{E}^{\prime}=(0,0,2 \alpha) and B^(')=(B_(x)^('),alpha//c,B_(z)^('))\mathbf{B}^{\prime}=\left(B_{x}^{\prime}, \alpha / c, B_{z}^{\prime}\right). Determine B_(x)^(')B_{x}^{\prime} and B_(z)^(')B_{z}^{\prime}.
Problem 1.126 Observer AA measures the electric and magnetic field strengths to be E=(0,beta,-beta)\mathbf{E}=(0, \beta,-\beta) and B=(2beta//c,0,0)\mathbf{B}=(2 \beta / c, 0,0), respectively, where beta!=0\beta \neq 0. Another observer, observer BB, makes the same measurements and finds E^(')=(2beta,0,0)\mathbf{E}^{\prime}=(2 \beta, 0,0) and B^(')=(B_(x)^('),B_(y)^('),beta//c)\mathbf{B}^{\prime}=\left(B_{x}^{\prime}, B_{y}^{\prime}, \beta / c\right). Determine B_(x)^(')B_{x}^{\prime} and B_(y)^(')B_{y}^{\prime}.
Problem 1.127 Observer AA measures the electric and magnetic field strengths to be E=(alpha,0,0)\mathbf{E}=(\alpha, 0,0) and B=(alpha//c,0,2alpha//c)\mathbf{B}=(\alpha / c, 0,2 \alpha / c), respectively, where alpha!=0\alpha \neq 0. Another observer, observer BB, makes the same measurements and finds E^(')=(E_(x)^('),alpha,0)\mathbf{E}^{\prime}=\left(E_{x}^{\prime}, \alpha, 0\right) and B^(')=(alpha//c,B_(y)^('),alpha//c)\mathbf{B}^{\prime}=\left(\alpha / c, B_{y}^{\prime}, \alpha / c\right). Express E_(x)^(')E_{x}^{\prime} and B_(y)^(')B_{y}^{\prime} in terms of alpha\alpha and cc. Finally, a third observer, observer CC, is moving relative to observer BB with constant velocity vv along the positive xx-axis of observer BB. Find the electric and magnetic field strengths, E^('')\mathbf{E}^{\prime \prime} and B^('')\mathbf{B}^{\prime \prime}, as observer CC measures them.
Problem 1.128 Assume that a muon originally travels vertically down toward the ground from an altitude of 10 km . There is a magnetic field coming from the Earth of B=50 muTB=50 \mu \mathrm{~T} affecting the motion of the muon. To make a simple model we take the magnetic field to be constant all the way from 10 km altitude to ground level. Suppose the field lines go from south to north and we are in Japan on the northern hemisphere. How far in length and in which direction is the deviation from the point where the muon would hit the ground without magnetic field, compared to where it hits the ground due to the deviation induced by the magnetic field of the Earth, if it has the energy of 2 GeV and is negatively charged?
Hint: The combination cBc B, where cc is the speed of light, has the value cB=300V//mc B=300 \mathrm{~V} / \mathrm{m}, for B=1muTB=1 \mu \mathrm{~T}. The trajectory of a charged particle in a homogeneous magnetic field is a circle, it is sufficient to compute the radius of the circle and then use geometric arguments.
Problem 1.129 An observer in the system SS has observed an electromagnetic field tensor F^(mu nu)F^{\mu \nu} with nonvanishing E\boldsymbol{E} - and B\boldsymbol{B}-fields. Performing a Lorentz transformation with velocity uu along the positive x_(1)x_{1}-axis to another system S^(')S^{\prime} he finds that the BB-field is absent, i.e., all its components are equal to 0 . What is the electric field in this system expressed in uu and the components of the electric field in SS ?
Problem 1.130 In an inertial frame SS there is a constant time-independent magnetic field BB and no electric field (E=0)(\boldsymbol{E}=\mathbf{0}). Consider another inertial frame S^(')S^{\prime}, which moves with velocity vv along the positive x^(1)x^{1}-axis of SS.
a) What are the E^(')\boldsymbol{E}^{\prime} and B^(')\boldsymbol{B}^{\prime} fields in the system S^(')S^{\prime} expressed in the original B\boldsymbol{B}-field and the velocity vv ?
b) Verify that the Lorentz invariants are indeed invariant under this transformation.
Problem 1.131 An electron in a linear particle accelerator of length L=3kmL=3 \mathrm{~km} (e.g., SLAC in California, USA) is accelerated through an electric potential UU.
a) Compute the trajectory x(t)x(t) of this electron for 0 < |x(t)| < L0<|x(t)|<L if its motion starts at time t=0t=0 at rest at one end of the accelerator.
b) Compute the time it takes for this electron to pass through the whole accelerator.
c) Compute the time dependence of the energy of this electron in the accelerator.
Problem 1.132 a) Find the electric and magnetic fields E\boldsymbol{E} and BB generated by a particle with charge qq moving with constant velocity vv parallel with the xx-axis in an
inertial system SS, using that the electric and magnetic potentials in the particle's rest frame are
we use the notation A=(A^(1),A^(2),A^(3))\boldsymbol{A}=\left(A^{1}, A^{2}, A^{3}\right), and similarly for E,B\boldsymbol{E}, \boldsymbol{B}, and x\boldsymbol{x}.
b) Explain why it is possible to check your result in a) by computing E*B\boldsymbol{E} \cdot \boldsymbol{B} and E^(2)-B^(2)\boldsymbol{E}^{2}-\boldsymbol{B}^{2} in both inertial systems. Perform these checks!
Problem 1.133 Bubble chambers were frequently used in the 1960s in particle collision experiments. In a bubble chamber, there is a strong constant magnetic field, which bends the motion of charged particles. The charged particles give rise to bubbles, which make the trajectories of the charged particles visible.
a) In the lab frame of the bubble chamber, there is a strong magnetic field in the zz-direction and no electric field. Use the Lorentz force law to show that the trajectory of a charge particle can be parametrized in the lab frame as
{:(1.17)x=R cos omega tau","quad y=-R sin omega tau",":}\begin{equation*}
x=R \cos \omega \tau, \quad y=-R \sin \omega \tau, \tag{1.17}
\end{equation*}
and determine omega\omega. Show that for a charged particle, you can obtain the 3-momentum from knowing the radius of the trajectory and the strength of the magnetic field (any energy losses can be neglect)
where Sigma^(-)\Sigma^{-}and pi^(-)\pi^{-}are known charged particles. Here X^(0)X^{0} is an unknown uncharged particle, which we cannot see, since it does not give rise to bubbles. For the other two particles, we know their rest masses and their trajectory radii R_(Sigma)R_{\Sigma} and R_(pi)R_{\pi} (therefore, we also know their 3-momenta). From this information, derive an expression for the rest mass of the unknown particle expressed in terms of the rest masses M_(Sigma)M_{\Sigma} of Sigma^(-)\Sigma^{-} and M_(pi)M_{\pi} of pi^(-)\pi^{-}, and their respective 3-momenta, as well as the angle theta\theta between the recorded trajectories of the charged particles close to the collision.
Problem 1.134 Starting from the plane wave solution to Maxwell's equations
show that the electric and magnetic fields are orthogonal and have the same magnitude without referring to a particular gauge condition.
Problem 1.135 Assume that the electromagnetic field in an inertial frame SS satisfies |E|=|B||\boldsymbol{E}|=|\boldsymbol{B}| and that the angle between the electric and magnetic field is alpha\alpha. In another inertial frame, the fields are E^(')\boldsymbol{E}^{\prime} and B^(')\boldsymbol{B}^{\prime} with a corresponding angle alpha^(')\alpha^{\prime}. Show that
where r=sqrt(x^(2)+y^(2)+z^(2))r=\sqrt{x^{2}+y^{2}+z^{2}} is the distance to the particle. Compute the electromagnetic stress-energy tensor T^(mu v)T^{\mu v} in (x,y,z)=(1,0,0)(x, y, z)=(1,0,0) and the corresponding trace T_(mu)^(mu)T_{\mu}^{\mu}.
Problem 1.137 Starting from Maxwell's equations and without assuming a particular gauge condition, show that the components of the electromagnetic field tensor F^(mu v)F^{\mu v} satisfy the sourced wave equation
and express the source tensor S^(mu nu)S^{\mu \nu} in terms of the 4-current density J^(mu)J^{\mu}.
Problem 1.138 The electromagnetic stress-energy tensor is given by
express T_(mu)^(v)T_{\mu}^{v} in terms of the 4 -vector kk. You also need to assure that the wave actually fulfills Maxwell's equations in the absence of a source term del_(mu)F^(mu nu)=0\partial_{\mu} F^{\mu \nu}=0.
Hint: You may assume the Lorenz gauge condition del_(mu)A^(mu)=0\partial_{\mu} A^{\mu}=0.
Problem 1.139 The 4-potential A_(mu)A_{\mu} is not physical, but may be transformed according to A_(mu)|->A_(mu)+del_(mu)varphiA_{\mu} \mapsto A_{\mu}+\partial_{\mu} \varphi, where varphi\varphi is a scalar field, without changing the physical observables. Show that the physical electromagnetic field tensor F^(mu nu)F^{\mu \nu} is invariant under this transformation.
Problem 1.140 The electric field of an electric dipole with dipole moment d=de_(z)d=d e_{z} is given by
{:(1.25)E=(d)/(4piepsi_(0))((3xz)/(r^(5))e_(x)+(3yz)/(r^(5))e_(y)+(3z^(2)-r^(2))/(r^(5))e_(z)):}\begin{equation*}
\boldsymbol{E}=\frac{d}{4 \pi \varepsilon_{0}}\left(\frac{3 x z}{r^{5}} \boldsymbol{e}_{x}+\frac{3 y z}{r^{5}} \boldsymbol{e}_{y}+\frac{3 z^{2}-r^{2}}{r^{5}} \boldsymbol{e}_{z}\right) \tag{1.25}
\end{equation*}
in its rest frame SS. Compute the value of the quantity F_(mu nu) tilde(F)^(mu nu)=epsi^(mu nu sigma rho)F_(mu nu)F_(sigma rho)F_{\mu \nu} \tilde{F}^{\mu \nu}=\varepsilon^{\mu \nu \sigma \rho} F_{\mu \nu} F_{\sigma \rho} as a function of time and position in the frame S^(')S^{\prime}, which is moving in the positive xx-direction with velocity vv relative to S^(')S^{\prime}.
Problem 1.141 A particle at rest acting as an electric monopole and a magnetic dipole has the electromagnetic fields
Figure 1.10 Two particles (each with charge qq ) and the plane SS equidistant from both particles.
where rr is the position vector relative to the particle, rr its magnitude, qq the charge of the particle, and mm its magnetic dipole moment. Determine whether or not there exists a region of spacetime where the electric field is equal to zero in some inertial frame (although that frame may generally be different for different points in the region). If such a region exists, determine the shape of the region in the particle's rest frame.
Problem 1.142 Two particles with the same charge qq are held fixed with a separation distance dd (see Figure 1.10). Compute the stress-energy tensor of the static electric field between the charges and use your result to find the total 4 -force between the electromagnetic fields on either side of the plane SS that is equidistant from both charges.
Problem 1.143 Compute the Lorentz 4 -force between two electrons moving in parallel with constant velocity v\boldsymbol{v} and a separation dd orthogonal to the direction of motion.
1.8 Energy-Momentum Tensor
Problem 1.144 Determine the momentum density of a gas consisting of massless noninteracting particles in a frame which is moving with velocity v\boldsymbol{v} relative to the gas rest frame. Express your result in terms of vv and the energy density of the gas in the frame where the gas is moving.
Problem 1.145 A star cruiser is moving through space with velocity vv relative to the galaxy. Suddenly it encounters a gas cloud of dust particles. What is the 4 -force from the dust cloud on the star cruiser at the moment it enters the cloud? You may assume that the star cruiser has a cross sectional area AA relative to the direction of
motion and that all of the dust particles encountered will be absorbed in the hull of the star cruiser. In addition to computing the 4 -force, motivate and state whether it is pure, heatlike, or neither.
Problem 1.146 The energy-momentum tensor of a string with tension tt is given by
where rho_(0)\rho_{0} is the string density and sigma=t//A < rho_(0)\sigma=t / A<\rho_{0} is the stress across the string cross section AA.
a) Does a frame exist where the stress (T^(11))\left(T^{11}\right) is equal to zero?
b) Does a frame exist where the energy density is smaller than rho_(0)\rho_{0} ?
Your answers should be accompanied by solid argumentation.
Problem 1.147 A pure photon gas such as the cosmic microwave background (CMB) can be described as a perfect fluid with pressure p=rho_(0)//3p=\rho_{0} / 3 in its rest frame. In a frame moving with velocity vv in relation to the rest frame of the CMB, compute the energy density, momentum density, and stress tensors. In addition, comment on whether the shear stress (off-diagonal elements of the stress tensor) in an arbitrary frame is zero or not.
Problem 1.148 In a perfect fluid with proper density rho_(0)\rho_{0} and positive proper pressure pp, find an expression for the energy density rho\rho in an arbitrary inertial frame S^(')S^{\prime} and derive an upper bound on rho//gamma^(2)\rho / \gamma^{2}, where gamma\gamma is the gamma factor between the fluid's rest frame and S^(')S^{\prime}.
Problem 1.149 The energy density in the frame of an observer with 4-velocity VV is given by rho=T_(mu nu)V^(mu)V^(nu)\rho=T_{\mu \nu} V^{\mu} V^{\nu}. The weak energy condition is a condition requiring the energy density to be nonnegative for all observers, i.e., rho >= 0\rho \geq 0. For a perfect fluid, determine the condition on the equation of state parameter ww in the relation p=wrho_(0)p=w \rho_{0} that the weak energy condition implies.
1.9 Lagrange's Formalism
Problem 1.150 The 4-momentum of a free particle of mass mm is p^(mu)=mcx^(˙)^(mu)p^{\mu}=m c \dot{x}^{\mu}.
a) Show that the momentum is conserved (i.e., independent of time) by deriving the Euler-Lagrange variational equations for the Lagrangian L=p^(2)//(2m)\mathscr{L}=p^{2} /(2 m) in Minkowski space.
b) When the particle moves in an electromagnetic field, one can obtain the relevant equations of motion by using the substitution p|->p+qA//cp \mapsto p+q A / c, where A=A(x)A=A(x) is the electromagnetic potential and qq is the charge of the particle. Show that, to lowest nontrivial order in qq, the equations of motion for the particle give the equations of the Lorentz force.
General Relativity Theory
2.1 Some Differential Geometry
Problem 2.1 Show that the function f(x,y)=x^(2)+yf(x, y)=x^{2}+y is a smooth function on the unit sphere S^(2)subR^(3)\mathbb{S}^{2} \subset \mathbb{R}^{3}.
Problem 2.2 On the unit sphere M=S^(2)M=\mathbb{S}^{2}, we use the spherical coordinates theta\theta and phi\phi, except at the poles theta=0,pi\theta=0, \pi. A curve can then be parametrized as (theta(s),phi(s))(\theta(s), \phi(s)). A tangent vector v inT_(p)S^(2)v \in T_{p} \mathbb{S}^{2} is given by its components v=(v_(theta),v_(phi))v=\left(v_{\theta}, v_{\phi}\right) with v_(theta)=theta^(˙)(s_(0))v_{\theta}=\dot{\theta}\left(s_{0}\right), v_(phi)=phi(s_(0))v_{\phi}=\phi\left(s_{0}\right), and p=(theta(s_(0)),phi(s_(0)))p=\left(\theta\left(s_{0}\right), \phi\left(s_{0}\right)\right). How would you describe a tangent vector at the poles theta=0,pi\theta=0, \pi ?
Problem 2.3 Find the metric tensor and the Christoffel symbols in the twodimensional Euclidean plane in the following coordinates
a) ss and tt defined by x=se^(t)x=s e^{t} and y=se^(-t)y=s e^{-t}.
b) uu and vv defined by x=ux=u and y=v^(2)y=v^{2}.
In both cases, discuss where in the Euclidean plane the new coordinates provide a well-defined coordinate system.
Problem 2.4 Let alpha(t)\alpha(t) and beta(t)\beta(t) be a pair of smooth curves on a manifold MM such that alpha(t_(0))=beta(t_(0))\alpha\left(t_{0}\right)=\beta\left(t_{0}\right). Show that the condition
{:(2.1)(d)/(dt)x^(i)(alpha(t))|_(t=t_(0))=(d)/(dt)x^(i)(beta(t))|_(t=t_(0))quad" for "i=1","2","dots","n",":}\begin{equation*}
\left.\frac{d}{d t} x^{i}(\alpha(t))\right|_{t=t_{0}}=\left.\frac{d}{d t} x^{i}(\beta(t))\right|_{t=t_{0}} \quad \text { for } i=1,2, \ldots, n, \tag{2.1}
\end{equation*}
is independent of the choice of local coordinates x^(i)x^{i}, i.e., if the curves are tangential in one coordinate system, then they are tangential in any other coordinate system.
Problem 2.5 Show that the system of first-order ordinary differential equations
defining the parallel transport along a curve on a manifold MM is coordinate independent in the sense that if the system is valid in one coordinate system, then it is also valid in any other coordinate system.
Problem 2.6 The distance between two points aa and bb on the unit sphere S^(2)\mathbb{S}^{2} along a curve gamma(s)=(theta(s),phi(s))\gamma(s)=(\theta(s), \phi(s)) is given by
{:[ℓ[gamma]-=int_(a)^(b)sqrt(g_(gamma(s))(gamma^(˙)(s),gamma^(˙)(s)))ds=int_(a)^(b)sqrt(g_(theta theta)theta^(˙)(s)^(2)+g_(phi phi)phi^(˙)(s)^(2))ds],[(2.3)=int_(a)^(b)sqrt(theta^(˙)(s)^(2)+sin^(2)theta(s)phi^(˙)(s)^(2))ds]:}\begin{align*}
\ell[\gamma] & \equiv \int_{a}^{b} \sqrt{g_{\gamma(s)}(\dot{\gamma}(s), \dot{\gamma}(s))} d s=\int_{a}^{b} \sqrt{g_{\theta \theta} \dot{\theta}(s)^{2}+g_{\phi \phi} \dot{\phi}(s)^{2}} d s \\
& =\int_{a}^{b} \sqrt{\dot{\theta}(s)^{2}+\sin ^{2} \theta(s) \dot{\phi}(s)^{2}} d s \tag{2.3}
\end{align*}
Use Euler-Lagrange equations to derive the geodesic equations on S^(2)\mathbb{S}^{2}.
Problem 2.7 Compute the Christoffel symbols on the unit sphere S^(2)\mathbb{S}^{2} with metric given by ds^(2)=dtheta^(2)+sin^(2)theta dphi^(2)d s^{2}=d \theta^{2}+\sin ^{2} \theta d \phi^{2}
a) directly from the metric.
b) using the general formula for the geodesic equations.
Problem 2.8 We define the Christoffel symbols on the unit sphere S^(2)\mathbb{S}^{2}, using spherical coordinates (theta,phi)(\theta, \phi). When theta!=0,pi\theta \neq 0, \pi, we find (see Problem 2.7)
and all other Gamma\Gamma are equal to zero. Show that the apparent singularity at theta=0,pi\theta=0, \pi can be removed by a better choice of coordinates at the poles of the sphere. Thus, the above affine connection extends to the whole S^(2)\mathbb{S}^{2}.
Problem 2.9 a) Let M=S^(2)M=\mathbb{S}^{2} and Gamma\Gamma be the affine connection in Problem 2.8. The coordinates theta(s)\theta(s) and phi(s)\phi(s) of a geodesic then satisfy the geodesic equations, i.e.,
Find the general solution to the geodesic equations.
b) Let MM and Gamma\Gamma be as in a). Furthermore, let (theta,phi)=(alpha s+beta,phi_(0))(\theta, \phi)=\left(\alpha s+\beta, \phi_{0}\right), where ss is the curve parameter and alpha,beta\alpha, \beta, and phi_(0)\phi_{0} are constants. Determine the parallel transport equations for a vector field X=(X^(theta),X^(phi))X=\left(X^{\theta}, X^{\phi}\right) and solve this set of equations. In addition, if uu is the tangent vector (1,1)(1,1) at the point (theta,phi)=((pi)/(4),0)(\theta, \phi)=\left(\frac{\pi}{4}, 0\right), then determine the parallel transported vector vv at the point (theta,phi)=((pi)/(2),0)(\theta, \phi)=\left(\frac{\pi}{2}, 0\right).
Problem 2.10 Determine the shortest path on the conical surface r=-azr=-a z which connects the points z=-h,varphi=0z=-h, \varphi=0 and z=-h,varphi=pi//2z=-h, \varphi=\pi / 2, where (r,varphi,z)(r, \varphi, z) are cylindrical coordinates and a > 0a>0 and h > 0h>0 constants.
Problem 2.11 A ship starts from a position in the Atlantic Ocean with coordinates 10^(@)N30^(@)W10^{\circ} \mathrm{N} 30^{\circ} \mathrm{W} (Cape Verde). It sails directly to the north to the 45^(@)45^{\circ} northern latitude (Azores, Portugal) and then it turns abruptly to the west and sails until it hits the 60^(@)60^{\circ} western longitude (Nova Scotia, Canada). Suppose a vector is parallel transported along the route of the ship (with the help of a gyroscope). Its initial direction is 45^(@)45^{\circ} (north-east). What is its final direction?
Problem 2.12 A vector is first parallel transported along a great circle on a sphere from a point AA on the equator to the North Pole NN, then again along a great circle from NN to another point BB on the equator, and finally, along the equator back to the point AA. Use the standard Riemannian metric on the sphere and prove that the vector is rotated in the above process by an angle theta\theta, which is directly proportional to the area of the geodesic triangle ANBA N B.
Problem 2.13 Let M=S^(2)subR^(3)M=\mathbb{S}^{2} \subset \mathbb{R}^{3}. Determine the metric gg on MM in terms of the spherical coordinates theta\theta and phi\phi. In particular, compute the inner product of the vectors (1,2)(1,2) and (2,-1)(2,-1) at the point (theta,phi)(\theta, \phi).
Problem 2.14 Compute the Riemann curvature tensor RR of the unit sphere S^(2)\mathbb{S}^{2}.
Problem 2.15 Consider the vector fields
in the xyx y-plane.
a) Determine the commutator [X,Y][X, Y].
b) Assume that an affine connection in the plane satisfies grad_(X)X=-Y,grad_(Y)Y=Y\nabla_{X} X=-Y, \nabla_{Y} Y=Y, grad_(Y)X=X\nabla_{Y} X=X, and that the torsion tensor TT vanishes. Compute the Riemann curvature tensor RR.
Problem 2.16 Let x^(1)x^{1} and x^(2)x^{2} be a pair of local coordinates and
Compute the components R^(1)_(1ij)R^{1}{ }_{1 i j} in the local coordinate basis, where i,j=1,2i, j=1,2, of the Riemann curvature tensor.
Problem 2.17 A manifold M\mathcal{M} of dimension 3 has a basis of orthonormal vector fields {L_(1),L_(2),L_(3)}\left\{L_{1}, L_{2}, L_{3}\right\} with commutation relations
{:(2.10)[L_(i),L_(j)]=epsilon_(ijk)L_(k)","quad" where "i","j","k=1","2","3.:}\begin{equation*}
\left[L_{i}, L_{j}\right]=\epsilon_{i j k} L_{k}, \quad \text { where } i, j, k=1,2,3 . \tag{2.10}
\end{equation*}
Determine the Levi-Civita connection grad_(i)=grad_(L_(i))(1 <= i <= 3)\nabla_{i}=\nabla_{L_{i}}(1 \leq i \leq 3) and its Riemann curvature tensor RR.
Hint: The Levi-Civita connection is the unique metric-compatible torsion-free connection. Use the symmetry properties of the Christoffel symbols coming from this, several times, to evaluate them.
Problem 2.18 Let xx and yy be local coordinates on a surface SS with x+y!=0x+y \neq 0. Define a metric tensor gg by g_(xx)=1,g_(xy)=g_(yx)=x+yg_{x x}=1, g_{x y}=g_{y x}=x+y, and g_(yy)=1+(x+y)^(2)g_{y y}=1+(x+y)^{2}. Let grad\nabla be an affine connection defined by
b) Consider the parallel transport of a pair of vectors starting from the point (x,y)=(1,1)(x, y)=(1,1), counterclockwise along the full circle with center at (x,y)=(2,2)(x, y)=(2,2) and radius r=sqrt2r=\sqrt{2}. Assume that the initial angle between the vectors is pi//3\pi / 3. What is the angle after the parallel transport around the loop?
Problem 2.19 Three ants are walking on a two-dimensional surface embedded in a flat three-dimensional Euclidean space as
Ant #3: r=lambda^(1//2),phi=ln lambda,quad lambda > 0r=\lambda^{1 / 2}, \phi=\ln \lambda, \quad \lambda>0.
a) Compute the induced metric on the two-dimensional surface.
b) Investigate if the ants are walking along geodesics or not.
Problem 2.20 Derive the explicit form of the geodesic equation on the hyperboloid x^(2)+y^(2)-z^(2)=a^(2)x^{2}+y^{2}-z^{2}=a^{2} with x,yx, y, and zz being Cartesian coordinates on the flat Euclidean three-dimensional space (i.e., the metric is ds^(2)=dx^(2)+dy^(2)+dz^(2)d s^{2}=d x^{2}+d y^{2}+d z^{2} ) and a > 0a>0 a constant. Using the coordinates rr and varphi\varphi such that x=r cos(varphi)x=r \cos (\varphi) and y=r sin(varphi)y=r \sin (\varphi), compute also all Christoffel symbols for this hyperboloid.
Problem 2.21 Consider the surface (ct)^(2)-x^(2)-y^(2)=-K^(2)(c t)^{2}-x^{2}-y^{2}=-K^{2} in the three-dimensional Minkowski space R^(3)\mathbb{R}^{3} with metric signature +-- .
a) Compute the metric tensor on the surface in a suitable coordinate system. Is it positive definite? If not, of what type?
b) Find the geodesic equations. In particular, find a geodesic starting from the point (0,0,K)(0,0, K) and going in the direction of the tangent vector (c,0,0)(c, 0,0).
Problem 2.22 Consider the pseudo-Riemannian metric
{:(2.17)ds^(2)=(dx^(1))^(2)+(dx^(2))^(2)-(dx^(3))^(2)-(dx^(4))^(2):}\begin{equation*}
d s^{2}=\left(d x^{1}\right)^{2}+\left(d x^{2}\right)^{2}-\left(d x^{3}\right)^{2}-\left(d x^{4}\right)^{2} \tag{2.17}
\end{equation*}
in R^(4)\mathbb{R}^{4}. This induces a pseudo-Riemannian metric gg on the surface
a) Show that the metric gg on SS is Lorentzian, i.e., it has one timelike and two spacelike directions at each point.
b) Construct a pair of constants of motions for freely falling bodies by integrating the geodesic equations on SS once.
Problem 2.23 The flow lines generated by a vector field XX are smooth curves gamma(t)\gamma(t) such that
along the curve. Assume that all flow lines for a vector field XX are geodesics with respect to a connection determined by the Christoffel symbols Gamma_(ij)^(k)\Gamma_{i j}^{k}. Derive a set of partial differential equations for the components of XX giving a necessary and sufficient condition for the above property of XX.
Problem 2.24 Parametrize the points on the spin group SU(2)\mathrm{SU}(2) as
where x=(x^(1),x^(2),x^(3))inR^(3),r=|x|\boldsymbol{x}=\left(x^{1}, x^{2}, x^{3}\right) \in \mathbb{R}^{3}, r=|\boldsymbol{x}| and the sigma_(k)\sigma_{k} 's are the Hermitian 2xx22 \times 2 Pauli matrices
We can identify the point g(x)g(x) as a point on the 3-dimensional unit sphere S^(3)subR^(4)\mathbb{S}^{3} \subset \mathbb{R}^{4}, the first coordinate is cos(r)\cos (r) and the remaining three coordinates are sin(r)x//r\sin (r) \boldsymbol{x} / r. Show that the 1-parameter subgroups t|->e^("ita "*sigma)t \mapsto e^{\text {ita } \cdot \sigma}, where a inR^(3)\boldsymbol{a} \in \mathbb{R}^{3}, are geodesics with respect to the standard metric on S^(3)\mathbb{S}^{3} coming from the Euclidean metric in R^(4)\mathbb{R}^{4}.
Hint: It is more convenient to use the Euler-Lagrange equations coming from the metric element (derive the formula!) in terms of the angular coordinates theta,phi\theta, \phi of the vector x inR^(3)\boldsymbol{x} \in \mathbb{R}^{3} and the radial coordinate rr.
Problem 2.25 a) Derive the relation between the Christoffel symbols Gamma_(mu nu)^(lambda)\Gamma_{\mu \nu}^{\lambda} and the metric tensor g_(mu nu)g_{\mu \nu} from the following conditions: (i) D_(lambda)g_(mu nu)=0D_{\lambda} g_{\mu \nu}=0, where D_(lambda)D_{\lambda} is the covariant derivative, and (ii) Gamma_(mu nu)^(lambda)=Gamma_(nu mu)^(lambda)\Gamma_{\mu \nu}^{\lambda}=\Gamma_{\nu \mu}^{\lambda}. (The result is the so-called "fundamental theorem" in Riemannian geometry.)
b) Consider the vector field (V^(mu))=(x,-t)\left(V^{\mu}\right)=(x,-t), i.e., V^(0)=xV^{0}=x and V^(1)=-tV^{1}=-t, in twodimensional Minkowski spacetime with coordinates (x^(mu))=(t,x)\left(x^{\mu}\right)=(t, x) and metric ds^(2)=d s^{2}=dt^(2)-dx^(2)d t^{2}-d x^{2}. Compute all components of the tensor T_(mu)^(nu)=D_(mu)V^(v)T_{\mu}{ }^{\nu}=D_{\mu} V^{v} in this coordinate system. Compute also the component T_(0)^(1)T_{0}{ }^{1} of this tensor in Rindler coordinates (x^('mu))=(lambda,a)\left(x^{\prime \mu}\right)=(\lambda, a) defined as
Problem 2.26 a) Write the transformation law for a tensor with components S^(mu nu)S^{\mu \nu} under a general coordinate transformation x^(mu)|->x^('mu)x^{\mu} \mapsto x^{\prime \mu}, i.e., give a general formula for S^('mu nu)S^{\prime \mu \nu}.
b) Write D_(mu)S^(mu v)D_{\mu} S^{\mu v} in terms of partial derivatives and Christoffel symbols.
c) Consider the tensor with components S^(12)=-S^(21)=2xyS^{12}=-S^{21}=2 x y and S^(11)=S^(22)=0S^{11}=S^{22}=0 on the two-dimensional plane with coordinates (x^(mu))=(x,y)\left(x^{\mu}\right)=(x, y) and metric ds^(2)=d s^{2}=dx^(2)+dy^(2)d x^{2}+d y^{2}. (i) Compute the components S^('mu nu)S^{\prime \mu \nu} of this tensor in polar coordinates (x^('mu))=(r,varphi)\left(x^{\prime \mu}\right)=(r, \varphi). (ii) Compute D_(mu)S^(mu nu)D_{\mu} S^{\mu \nu}.
Problem 2.27 Consider AdS_(2)\mathrm{AdS}_{2} which is a two-dimensional curved spacetime with coordinates (x^(mu))=(x^(0),x^(1))=(t,r)\left(x^{\mu}\right)=\left(x^{0}, x^{1}\right)=(t, r) and the metric given by
{:(2.23)ds^(2)=v[(r^(2)-1)dt^(2)-(1)/(r^(2)-1)dr^(2)]",":}\begin{equation*}
d s^{2}=v\left[\left(r^{2}-1\right) d t^{2}-\frac{1}{r^{2}-1} d r^{2}\right], \tag{2.23}
\end{equation*}
where v > 0v>0 is some constant. Consider also on AdS_(2)\mathrm{AdS}_{2} and in the coordinates x^(mu)x^{\mu}, a tensor field S_(mu nu)S_{\mu \nu} with the following components S_(00)=a(r^(2)-1),S_(11)=-a//(r^(2)-1)S_{00}=a\left(r^{2}-1\right), S_{11}=-a /\left(r^{2}-1\right), and S_(01)=S_(10)=0S_{01}=S_{10}=0 for some constant a > 0a>0.
a) Compute all Christoffel symbols for AdS_(2)\mathrm{AdS}_{2} in the coordinates x^(mu)x^{\mu}.
b) Compute S^(mu nu)S^{\mu \nu} and D_(mu)S^(mu nu)D_{\mu} S^{\mu \nu}.
c) Consider another coordinate system (x^('mu))=(x^('0),x^('1))=(theta,eta)\left(x^{\prime \mu}\right)=\left(x^{\prime 0}, x^{\prime 1}\right)=(\theta, \eta) on AdS_(2)\operatorname{AdS}_{2} defined by theta=at\theta=a t and r=cosh(eta)r=\cosh (\eta) with aa being the same constant as above. Transform the tensor S_(mu nu)S_{\mu \nu} to this new coordinate system, i.e., compute S_(mu nu)^(')S_{\mu \nu}^{\prime}.
Problem 2.28 Consider the so-called Rindler coordinate system (x^('mu))=\left(x^{\prime \mu}\right)=(x^('0),x^('1))=(lambda,a)\left(x^{\prime 0}, x^{\prime 1}\right)=(\lambda, a) in two-dimensional Minkowski space defined by
where (x^(mu))=(t,x)\left(x^{\mu}\right)=(t, x) are the usual coordinates, i.e., ds^(2)=dt^(2)-dx^(2)d s^{2}=d t^{2}-d x^{2}.
a) Find the metric tensor and the Christoffel symbols in this coordinate system.
b) Determine expressions for the divergence of a vector field and the Laplacian of a scalar field in two-dimensional Minkowski space in Rindler coordinates.
c) A tensor of rank two on two-dimensional Minkowski space has the following components in the coordinates (x^(mu))=(t,x):T_(0)^(0)=-T_(1)^(1)=x^(2)-t^(2)\left(x^{\mu}\right)=(t, x): T_{0}^{0}=-T_{1}^{1}=x^{2}-t^{2} and T_(0)^(1)=T_{0}^{1}=T_(1)^(0)=0T_{1}^{0}=0. Compute the component T^('0)_(0)T^{\prime 0}{ }_{0}, i.e., T^('mu)_(v)T^{\prime \mu}{ }_{v} for mu=v=0\mu=v=0, of this tensor in Rindler coordinates.
Problem 2.29 Consider the metric
{:(2.25)ds^(2)=dt^(2)-dr^(2)-r^(2)dphi^(2):}\begin{equation*}
d s^{2}=d t^{2}-d r^{2}-r^{2} d \phi^{2} \tag{2.25}
\end{equation*}
Find expressions for the covariant derivative grad_(mu)V_(v)\nabla_{\mu} V_{v} and the divergence grad_(mu)V^(mu)\nabla_{\mu} V^{\mu}.
Problem 2.30 The parallel transport of a vector V^(mu)V^{\mu} along a curve parametrized by lambda\lambda is given by the condition
from parallel transporting a tangent vector.
b) If one allows the tangent vector T^(mu)T^{\mu} to change in size, one can generalize the condition (dx^(mu))/(d lambda)grad_(mu)V^(nu)=0\frac{d x^{\mu}}{d \lambda} \nabla_{\mu} V^{\nu}=0 to
{:(2.28)T^(mu)grad_(mu)T^(v)=alphaT^(v)","quad" where "T^(mu)=(dx^(mu))/(d lambda).:}\begin{equation*}
T^{\mu} \nabla_{\mu} T^{v}=\alpha T^{v}, \quad \text { where } T^{\mu}=\frac{d x^{\mu}}{d \lambda} . \tag{2.28}
\end{equation*}
Show that one can get back the original condition, where one has zero on the righthand side, by making a reparametrization lambda rarr tau(lambda)\lambda \rightarrow \tau(\lambda). Show also how the geodesic equation is modified by the extra term on the right-hand side.
Problem 2.31 A conformal transformation of the metric tensor is defined as
for an arbitrary positive function ff.
a) Show that a conformal transformation preserves the angle between any two vectors.
b) A null curve is a curve for which all tangent vectors are null vectors. Show that all null curves remain null curves after performing a conformal transformation.
Problem 2.32 A sphere is described locally by the two coordinates theta\theta and varphi\varphi is embedded in R^(3)\mathbb{R}^{3} according to
where RR and 0 < alpha < 10<\alpha<1 are constants (note that while the topology of this manifold is a sphere, this is not the standard embedding of the sphere in Euclidean space). Compute the induced metric tensor and the Christoffel symbols.
2.2 Christoffel Symbols, Riemann and Ricci Tensors, and Einstein's Equations
Problem 2.33 a) Let Gamma_(mu nu)^(lambda)\Gamma_{\mu \nu}^{\lambda} be the Levi-Civita connection associated to a metric tensor g_(mu nu)g_{\mu \nu}. Show that Gamma_(mu nu)^(mu)=(1)/(2) bar(g)^(-1)del_(nu) bar(g)\Gamma_{\mu \nu}^{\mu}=\frac{1}{2} \bar{g}^{-1} \partial_{\nu} \bar{g}, where bar(g)=det(g_(mu nu))\bar{g}=\operatorname{det}\left(g_{\mu \nu}\right).
b) Derive from the definition of covariant differentiation the transformation rule for the Christoffel symbols with respect to general coordinate transformations.
c) Show directly from the definition of parallel transport that in a parallel transport defined by the Levi-Civita connection, Gamma_(mu nu)^(lambda)=(1)/(2)g^(lambda omega)(del_(mu)g_(nu omega)+del_(nu)g_(mu omega)-del_(omega)g_(mu nu))\Gamma_{\mu \nu}^{\lambda}=\frac{1}{2} g^{\lambda \omega}\left(\partial_{\mu} g_{\nu \omega}+\partial_{\nu} g_{\mu \omega}-\partial_{\omega} g_{\mu \nu}\right), the length of a parallel transported vector is constant.
Problem 2.34 For any two vector fields U^(mu)U^{\mu} and V^(mu)V^{\mu} on a manifold with metric g_(mu nu)g_{\mu \nu} equipped with the Levi-Civita connection, show that if
a) Compute the Christoffel symbols Gamma_(theta i)^(j)\Gamma_{\theta i}^{j} and Gamma_(phi i)^(j)\Gamma_{\phi i}^{j} in the orthonormal basis e_(1)=del_(theta)e_{1}=\partial_{\theta}, e_(2)=(1)/(sin theta)del_(phi),grad_(theta)e_(i)=Gamma_(theta i)^(j)e_(j)e_{2}=\frac{1}{\sin \theta} \partial_{\phi}, \nabla_{\theta} e_{i}=\Gamma_{\theta i}^{j} e_{j}, and grad_(phi)e_(i)=Gamma_(phi i)^(j)e_(j)\nabla_{\phi} e_{i}=\Gamma_{\phi i}^{j} e_{j}.
b) Prove that the parallel transport of a vector u=u_(1)e_(1)+u_(2)e_(2)=((u_(1))/(u_(2)))u=u_{1} e_{1}+u_{2} e_{2}=\binom{u_{1}}{u_{2}} around a closed loop gamma(t)\gamma(t) on S^(2)\mathbb{S}^{2} is given by the operation u^(')=Ruu^{\prime}=R u, where RR is a rotation by an angle Omega\Omega equal to the area of the region bounded by the loop gamma\gamma.
Hint: First, write the solution as a line integral of the Christoffel symbols around the loop, and then, apply Green's formula in the plane.
Problem 2.36 The Christoffel symbols for the flat Euclidean metric in R^(3)\mathbb{R}^{3} vanish. Compute the Christoffel symbols in the spherical coordinates (r,theta,varphi)(r, \theta, \varphi).
Problem 2.37 A sphere can be projected onto a plane using stereographic projection. We use the metric
{:(2.34)ds^(2)=R^(2)(dtheta^(2)+sin^(2)theta dphi^(2)):}\begin{equation*}
d s^{2}=R^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) \tag{2.34}
\end{equation*}
for the sphere. The xx and yy coordinates of the plane can be expressed in terms of the spherical coordinates as
a) Express the metric of the sphere in terms of the xx and yy coordinates.
b) Compute the Christoffel symbols using the metric obtained in a).
c) Draw a picture illustrating the stereographic projection.
Problem 2.38 A two-dimensional hyperbolic subspace x^(2)+y^(2)-t^(2)=1,z=0x^{2}+y^{2}-t^{2}=1, z=0 is embedded into the four-dimensional Minkowski space.
a) Parametrize the surface using only two parameters.
b) Compute the induced metric on the subspace.
c) Compute the Christoffel symbols in the subspace.
Problem 2.39 a) Starting from the definition of the curvature tensor
{:(2.36)R(X","Y)Z=[grad_(X),grad_(Y)]Z-grad_([X,Y])Z:}\begin{equation*}
R(X, Y) Z=\left[\nabla_{X}, \nabla_{Y}\right] Z-\nabla_{[X, Y]} Z \tag{2.36}
\end{equation*}
derive the formula for the components R^(omega)_(mu nu lambda)R^{\omega}{ }_{\mu \nu \lambda} in terms of the Christoffel symbols. Prove the first Bianchi identity
{:(2.37)R^(omega)_(mu nu lambda)+R_(nu lambda mu)^(omega)+R_(lambda mu nu)^(omega)=0",":}\begin{equation*}
R^{\omega}{ }_{\mu \nu \lambda}+R_{\nu \lambda \mu}^{\omega}+R_{\lambda \mu \nu}^{\omega}=0, \tag{2.37}
\end{equation*}
in the case when the torsion T=0T=0.
b) Prove the second Bianchi identity
vanishes.
Motivate that the vanishing of the covariant derivative of T^(mu v)T^{\mu v} coincides with local energy-momentum conservation for flat spacetime.
Problem 2.40 Derive the formula relating the Riemann curvature tensor to the parallel transport around an infinitesimal parallelogram.
Problem 2.41 Fix a metric on the paraboloid z=x^(2)+y^(2)z=x^{2}+y^{2} induced by the standard Euclidean metric in R^(3)\mathbb{R}^{3}. Compute the components of the Riemann curvature tensor on the paraboloid.
Hint: Use polar coordinates in the xyx y-plane.
Problem 2.42 Consider the two-dimensional metric
{:(2.40)ds^(2)=r^(2)(dr^(2)+r^(2)dphi^(2)):}\begin{equation*}
d s^{2}=r^{2}\left(d r^{2}+r^{2} d \phi^{2}\right) \tag{2.40}
\end{equation*}
a) Calculate the component R^(r)_(phi r phi)R^{r}{ }_{\phi r \phi} of the Riemann tensor.
b) In flat Euclidean space, the relation between area and circumference of a circle is C^(2)=4pi AC^{2}=4 \pi A. What is the relation for a circle around the origin for the above metric? The area is given by the integral intsqrt(det(g))drd phi\int \sqrt{\operatorname{det}(g)} d r d \phi.
Problem 2.43 Let MM be a Lorentzian manifold of dimension n=3n=3. Assume that there is an orthogonal basis of vector fields X,Y,ZX, Y, Z such that
[X,Y]=-Z,[Y,Z]=X,[Z,X]=Y[X, Y]=-Z,[Y, Z]=X,[Z, X]=Y,
where gg is the metric tensor. Compute the Christoffel symbols of the Levi-Civita connection and the Riemann curvature tensor in this basis.
Hint: Use the symmetry properties of the Christoffel symbols coming from the torsion-free property of the connection together with grad_(X)g=grad_(Y)g=grad_(Z)g=0\nabla_{X} g=\nabla_{Y} g=\nabla_{Z} g=0.
Problem 2.44 a) Show that the Ricci tensor R_(mu nu)=R^(lambda)_(mu lambda nu)R_{\mu \nu}=R^{\lambda}{ }_{\mu \lambda \nu} (and thus also the Einstein tensor) is symmetric when the Riemann curvature tensor R^(alpha)_(mu beta v)R^{\alpha}{ }_{\mu \beta v} has been constructed from a metric.
b) Show that in two spacetime dimensions the tensor R_(mu nu)-kg_(mu nu)RR_{\mu \nu}-k g_{\mu \nu} R vanishes for some number kk. Determine kk.
c) Show that any metric in a 1+1-dimensional spacetime satisfies Einstein's equations in vacuum ( T_(mu nu)=0T_{\mu \nu}=0 ), i.e, G_(mu nu)=0G_{\mu \nu}=0.
Hint: Use the (anti)symmetries
{:(2.41)R_(alpha beta mu nu)=-R_(beta alpha mu nu)=-R_(alpha beta v mu)=R_(mu nu alpha beta)",":}\begin{equation*}
R_{\alpha \beta \mu \nu}=-R_{\beta \alpha \mu \nu}=-R_{\alpha \beta v \mu}=R_{\mu \nu \alpha \beta}, \tag{2.41}
\end{equation*}
of the Riemann curvature tensor.
Problem 2.45 Consider the two-dimensional curved spacetime with the metric given by
{:(2.42)ds^(2)=(1)/(y^(2))(dt^(2)-dy^(2)):}\begin{equation*}
d s^{2}=\frac{1}{y^{2}}\left(d t^{2}-d y^{2}\right) \tag{2.42}
\end{equation*}
in coordinates (x^(mu))=(x^(0),x^(1))=(t,y)\left(x^{\mu}\right)=\left(x^{0}, x^{1}\right)=(t, y) with t inRt \in \mathbb{R} and y >= 0y \geq 0.
a) Find the geodesic equations and the Christoffel symbols.
b) Compute the Riemann curvature tensor and the Ricci scalar.
Problem 2.46 Calculate the Christoffel symbols, the Riemann curvature tensor, the Ricci tensor, and the Ricci scalar for the metric
{:(2.43)ds^(2)=drho^(2)+(a^(2)+rho^(2))dphi^(2)",":}\begin{equation*}
d s^{2}=d \rho^{2}+\left(a^{2}+\rho^{2}\right) d \phi^{2}, \tag{2.43}
\end{equation*}
where a > 0a>0 is a constant and the coordinates rho\rho and phi\phi vary in the intervals -oo < rho < oo-\infty<\rho<\infty and 0 <= phi < 2pi0 \leq \phi<2 \pi, respectively.
Problem 2.47 Show by direct computation of the Riemann curvature tensor that the curvature of the Rindler space with coordinates lambda\lambda and aa and line element
{:(2.44)ds^(2)=a^(2)dlambda^(2)-da^(2):}\begin{equation*}
d s^{2}=a^{2} d \lambda^{2}-d a^{2} \tag{2.44}
\end{equation*}
is zero.
Problem 2.48 Consider the curved two-dimensional spacetime t^(2)-x^(2)-y^(2)=-1t^{2}-x^{2}-y^{2}=-1 embedded in three-dimensional Minkowski spacetime with coordinates (x^(mu))=\left(x^{\mu}\right)=(t,x,y)(t, x, y) and the metric ds^(2)=dt^(2)-dx^(2)-dy^(2)d s^{2}=d t^{2}-d x^{2}-d y^{2}. Compute the metric tensor g_(mu nu)g_{\mu \nu} and the Ricci tensor R_(mu nu)R_{\mu \nu} for this two-dimensional spacetime and thus prove that
for some constant Lambda\Lambda to be determined. Perform this computation in the coordinate system (x^(mu))=(lambda,varphi)\left(x^{\mu}\right)=(\lambda, \varphi), where t=sinh(lambda),x=cosh(lambda)cos(varphi)t=\sinh (\lambda), x=\cosh (\lambda) \cos (\varphi), and y=y=cosh(lambda)sin(varphi)\cosh (\lambda) \sin (\varphi).
Problem 2.49 Compute the Ricci tensor for the two-dimensional spacetime AdS_(2)\mathrm{AdS}_{2} and in the coordinates x^(mu)x^{\mu} as defined in Problem 2.27.
Problem 2.50 Consider the 2-dimensional manifold MM defined by being the surface t^(2)+u^(2)-x^(2)=alpha^(2)t^{2}+u^{2}-x^{2}=\alpha^{2} (with alpha > 0\alpha>0 being a constant) embedded in a 3 -dimensional flat manifold with coordinates t,ut, u, and xx, and line element ds^(2)=dt^(2)+du^(2)-dx^(2)d s^{2}=d t^{2}+d u^{2}-d x^{2}.
a) Introduce suitable coordinates on MM and compute the line element for MM in terms of those coordinates.
b) Compute the Christoffel symbols in MM in the coordinates introduced in a).
c) Compute the Ricci scalar in MM.
Problem 2.51 a) Derive the geodesic equations and determine the metric tensor, the Christoffel symbols, the Riemann curvature tensor, the Ricci tensor, and the Ricci scalar for the spherically symmetric metric
{:[" 2.3 Maxwell's Equations and Energy-Momentum Tensor "],[(2.46)ds^(2)=g_(tt)(t","r)c^(2)dt^(2)+g_(rr)(t","r)dr^(2)-r^(2)(dtheta^(2)+sin^(2)theta dphi^(2))","]:}\begin{align*}
& \text { 2.3 Maxwell's Equations and Energy-Momentum Tensor } \\
& d s^{2}=g_{t t}(t, r) c^{2} d t^{2}+g_{r r}(t, r) d r^{2}-r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right), \tag{2.46}
\end{align*}
with the arbitrary functions v=nu(t,r)v=\nu(t, r) and rho=rho(t,r)\rho=\rho(t, r).
b) Derive the so-called Schwarzschild solution to Einstein's equations in empty space, i.e., solve G_(alpha beta)=0G_{\alpha \beta}=0 with the spherically symmetric metric given in a).
Hint: Assume that Birkhoff's theorem holds, which states that any spherically symmetric solution to G_(alpha beta)=0G_{\alpha \beta}=0 must be static (and asymptotically flat), i.e., v^(@)=0\stackrel{\circ}{v}=0 and rho^(@)=0\stackrel{\circ}{\rho}=0, where a circle denotes partial differentiation with respect to time tt.
Problem 2.52 Prove Birkhoff's theorem, i.e., prove that any spherically symmetric solution to Einstein's equations in empty space must be static.
2.3 Maxwell's Equations and Energy-Momentum Tensor
Problem 2.53 The energy-momentum tensor associated with the electromagnetic field strength tensor F^(mu v)F^{\mu v} is
where g^(mu nu)g^{\mu \nu} is the inverse of the metric tensor g_(mu nu)g_{\mu \nu}. Maxwell's equations in general relativity are written as in Minkowski space, except that partial derivatives are replaced by covariant derivatives, i.e., grad_(mu)F^(mu nu)=J^(nu)\nabla_{\mu} F^{\mu \nu}=J^{\nu}. Show that
Note that this does not violate the relation grad_(mu)T_("tot ")^(mu nu)=0\nabla_{\mu} T_{\text {tot }}^{\mu \nu}=0, since the T^(mu v)T^{\mu v} considered in this problem is just the electromagnetic part of the total energy-momentum tensor.
Problem 2.54 Show that half of Maxwell's equations, i.e.,
can be written precisely in the same form in general relativity; the equations transform covariantly in general coordinate transformations. Why is it unnecessary to write
Problem 2.55 Show that the covariant form grad_(mu)j^(mu)=0\nabla_{\mu} j^{\mu}=0 of the current conservation law can be written as bar(g)^(-(1)/(2))del_(mu)( bar(g)^((1)/(2))j^(mu))=0\bar{g}^{-\frac{1}{2}} \partial_{\mu}\left(\bar{g}^{\frac{1}{2}} j^{\mu}\right)=0, where bar(g)=-det(g_(mu nu));g_(mu nu)\bar{g}=-\operatorname{det}\left(g_{\mu \nu}\right) ; g_{\mu \nu} is a Lorentzian metric. Show that this is compatible with the generally covariant form grad_(mu)F^(mu nu)=j^(nu)\nabla_{\mu} F^{\mu \nu}=j^{\nu} of Maxwell's equations.
Problem 2.56 Assume that in a three-dimensional spacetime, there is a basis of vector fields {X_(0),X_(1),X_(2)}\left\{X_{0}, X_{1}, X_{2}\right\} with orthogonality relations g(X_(mu),X_(nu))=0g\left(X_{\mu}, X_{\nu}\right)=0 for mu!=v\mu \neq v and g(X_(0),X_(0))=-g(X_(1),X_(1))=-g(X_(2),X_(2))=1g\left(X_{0}, X_{0}\right)=-g\left(X_{1}, X_{1}\right)=-g\left(X_{2}, X_{2}\right)=1. In this basis (which is not a coordinate basis!), we define an affine connection by
We also assume that [X_(0),X_(1)]=X_(2),[X_(1),X_(2)]=-X_(0)\left[X_{0}, X_{1}\right]=X_{2},\left[X_{1}, X_{2}\right]=-X_{0}, and [X_(2),X_(0)]=X_(1)\left[X_{2}, X_{0}\right]=X_{1}.
a) Show that grad\nabla is the Levi-Civita connection associated to the metric gg.
b) Compute the energy-momentum tensor T_(mu nu)T_{\mu \nu} corresponding to the metric gg from Einstein's equations in the above basis.
Problem 2.57 The action for a point particle of mass MM in a curved spacetime is given by
where tau\tau is the proper time and gamma^(˙)\dot{\gamma} the 4 -velocity of the particle. What is the energymomentum tensor corresponding to such a point particle? Check that your expression takes the expected form in the case of standard coordinates in Minkowski space.
Problem 2.58 The Lagrangian density for an electromagnetic field A_(mu)A_{\mu} is given by
where F_(mu nu)=del_(mu)A_(nu)-del_(nu)A_(mu)F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu} and J^(mu)J^{\mu} is an external 4-current density that does not depend on A_(mu)A_{\mu}. Use this to derive the equations of motion for an electromagnetic field in a general spacetime.
Problem 2.59 The Lagrangian for a free massive scalar field phi\phi in a general spacetime is given by
where V(phi)V(\phi) is the potential density, which is a function of phi\phi only (i.e., it does not depend on the metric). Compute the components of the stress-energy tensor T_(mu nu)T_{\mu \nu}
and then simplify your expression in the case where g^(mu nu)(del_(mu)phi)(del_(nu)phi)g^{\mu \nu}\left(\partial_{\mu} \phi\right)\left(\partial_{\nu} \phi\right) is negligible compared to V(phi)V(\phi).
Problem 2.61 Consider the Robertson-Walker spacetime with metric
{:(2.59)ds^(2)=dt^(2)-cosh^(2)(Ht)dx^(2):}\begin{equation*}
d s^{2}=d t^{2}-\cosh ^{2}(H t) d x^{2} \tag{2.59}
\end{equation*}
where dx^(2)d x^{2} is the standard Euclidean line element in three dimensions. Show that this spacetime is not a vacuum solution to Einstein's field equations.
Problem 2.62 The Lagrangian density of a free electromagnetic field A_(mu)A_{\mu} is given by
i.e., the derivative of WW with respect to the curve parameter ss vanishes.
Hint: Use the symmetry of the Christoffel symbols from the Levi-Civita connection.
Problem 2.64 The symmetries of a spacetime metric are associated to so-called Killing vector fields. A vector field X is a Killing vector field if L_(X)g=0\mathcal{L}_{X} g=0; this means that
c) Show that the vector fields X_(mu)X_{\mu} in Problem 2.56 are all Killing vector fields.
Problem 2.65 Find the flows of the following vector fields and determine if they are Killing vector fields or not.
a) The field K=ydel_(x)-xdel_(y)K=y \partial_{x}-x \partial_{y} in the Euclidean plane with Cartesian coordinates xx and yy.
b) The field K=xdel_(t)-tdel_(x)K=x \partial_{t}-t \partial_{x} in two-dimensional Minkowski space with standard coordinates tt and xx.
Problem 2.66 Consider the paraboloid z=alpha(x^(2)+y^(2))z=\alpha\left(x^{2}+y^{2}\right) as a submanifold embedded in Euclidean three-dimensional space (R^(3))\left(\mathbb{R}^{3}\right).
a) Using coordinates rr and varphi\varphi such that x=r cos(varphi)x=r \cos (\varphi) and y=r sin(varphi)y=r \sin (\varphi), compute the induced metric tensor on the paraboloid.
b) Verify that the vector field K=del_(varphi)K=\partial_{\varphi} is a Killing vector field and find an expression for the corresponding conserved quantity for a geodesic in terms of the coordinates and their derivatives along the geodesic.
Problem 2.67 A torus can be parametrized using two angles theta\theta and varphi\varphi. The metric induced by a typical embedding in R^(3)\mathbb{R}^{3} corresponds to the line element
{:(2.66)ds^(2)=[R+rho sin(varphi)]^(2)dtheta^(2)+rho^(2)dvarphi^(2).:}\begin{equation*}
d s^{2}=[R+\rho \sin (\varphi)]^{2} d \theta^{2}+\rho^{2} d \varphi^{2} . \tag{2.66}
\end{equation*}
a) Find the Christoffel symbols corresponding to the Levi-Civita connection of this metric.
b) Find a Killing vector field for the torus and the corresponding conserved quantity along geodesics of the Levi-Civita connection.
c) It is possible to introduce a flat connection tilde(grad)\tilde{\nabla} with all connection coefficients tilde(Gamma)_(ab)^(c)=0\tilde{\Gamma}_{a b}^{c}=0 in the theta-varphi\theta-\varphi coordinate system. This connection is not metric compatible. Compute the components of the derivative tilde(grad)_(a)g\tilde{\nabla}_{a} g for this connection.
Problem 2.68 A wavy two-dimensional surface locally described by the coordinates rho\rho and varphi\varphi is embedded in R^(3)\mathbb{R}^{3} according to
where R_(0) > 0R_{0}>0 is a constant.
a) Compute the induced metric tensor and the Christoffel symbols.
b) Find the flow for each of the following vector fields and determine whether or not they are Killing vector fields:
Problem 2.69 The 2-dimensional de Sitter space dS_(2)\mathrm{dS}_{2} may be defined as the surface t^(2)-x^(2)-y^(2)=-r_(0)^(2)t^{2}-x^{2}-y^{2}=-r_{0}^{2} in 1+2-dimensional Minkowski space, where r_(0) > 0r_{0}>0 is a constant.
a) Introduce suitable coordinates on dS_(2)\mathrm{dS}_{2}.
b) Compute the components of the metric tensor induced by the embedding in Minkowski space in your selected coordinates.
c) Find (at least) two Killing vector fields on dS_(2)\mathrm{dS}_{2}.
2.5 Schwarzschild Metric
Problem 2.70 The Schwarzschild metric, when restricted to the plane theta=(pi)/(2)\theta=\frac{\pi}{2}, is given by
{:(2.69)ds^(2)=(1-(alpha )/(r))(dx^(0))^(2)-(1-(alpha )/(r))^(-1)dr^(2)-r^(2)dphi^(2):}\begin{equation*}
d s^{2}=\left(1-\frac{\alpha}{r}\right)\left(d x^{0}\right)^{2}-\left(1-\frac{\alpha}{r}\right)^{-1} d r^{2}-r^{2} d \phi^{2} \tag{2.69}
\end{equation*}
Derive the geodesic equations of motion for a test particle in this metric.
Problem 2.71 The Schwarzschild metric is normally written in terms of time and spherical coordinates. Transform this metric to coordinates (x^(1),x^(2),x^(3))=\left(x^{1}, x^{2}, x^{3}\right)=r(sin theta cos phi,sin theta sin phi,cos theta)r(\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta).
Problem 2.72 Consider the Schwarzschild metric ds^(2)=c^(2)dtau^(2)=(1-(2GM)/(c^(2)r))(dx^(0))^(2)-(1-(2GM)/(c^(2)r))^(-1)dr^(2)-r^(2)dOmega^(2)d s^{2}=c^{2} d \tau^{2}=\left(1-\frac{2 G M}{c^{2} r}\right)\left(d x^{0}\right)^{2}-\left(1-\frac{2 G M}{c^{2} r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2},
where tau\tau is the proper time, x^(0)=ctx^{0}=c t, and dOmega^(2)=dtheta^(2)+sin^(2)theta dphi^(2)d \Omega^{2}=d \theta^{2}+\sin ^{2} \theta d \phi^{2}.
a) Assuming circular motion in the equatorial plane, i.e., r=r_(0)r=r_{0}, where r_(0)r_{0} is a constant, derive Kepler's third law
where Delta t\Delta t is the period. Compare with the classical result.
b) Compute the proper time Delta tau\Delta \tau for one period of circular motion.
Problem 2.73 The Schwarzschild metric is given by
{:(2.72)ds^(2)=(1-(r_(**))/(r))(dx^(0))^(2)-(1-(r_(**))/(r))^(-1)dr^(2)-r^(2)dOmega^(2):}\begin{equation*}
d s^{2}=\left(1-\frac{r_{*}}{r}\right)\left(d x^{0}\right)^{2}-\left(1-\frac{r_{*}}{r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2} \tag{2.72}
\end{equation*}
where dOmega^(2)=dtheta^(2)+sin^(2)theta dphi^(2)d \Omega^{2}=d \theta^{2}+\sin ^{2} \theta d \phi^{2} and r_(**)-=2GM//c^(2)r_{*} \equiv 2 G M / c^{2} is the Schwarzschild radius. Find the worldlines for bodies, outside of the Schwarzschild horizon, radially freely falling toward the black hole.
Problem 2.74 Show that there are no circular free fall orbits inside of the radius r=3r_(**)//2r=3 r_{*} / 2 in the Scwharzschild spacetime.
Problem 2.75 For the Schwarzschild solution in the limit r≫r_(**)r \gg r_{*} and approximately circular orbits such that r=r_(0)+rhor=r_{0}+\rho, where rho≪r_(0)\rho \ll r_{0}, determine the ratio between the period of oscillations in rho\rho to the orbital period.
Problem 2.76 The optical size of a black hole is given by 4pib^(2)4 \pi b^{2}, where bb is the minimal impact parameter such that the past-null geodesics originate at r rarr oor \rightarrow \infty.
Find the optical size of the Schwarzschild black hole for which the line element is given by
{:(2.73)ds^(2)=(1-(R)/(r))dt^(2)-(1-(R)/(r))^(-1)dr^(2)-r^(2)dOmega^(2):}\begin{equation*}
d s^{2}=\left(1-\frac{R}{r}\right) d t^{2}-\left(1-\frac{R}{r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2} \tag{2.73}
\end{equation*}
Hint: For null geodesics in the Schwarzschild spacetime, the angular momentum is equal to the impact parameter (i.e., b=Lb=L ) for r^(˙)=1\dot{r}=1 at r rarr oor \rightarrow \infty.
Problem 2.77 Show that the Schwarzschild solution of Einstein's equation
{:(2.74)ds^(2)=(1-(r_(**))/(r))dt^(2)-(1-(r_(**))/(r))^(-1)dr^(2)-r^(2)[dtheta^(2)+sin(theta)^(2)dvarphi^(2)]",":}\begin{equation*}
d s^{2}=\left(1-\frac{r_{*}}{r}\right) d t^{2}-\left(1-\frac{r_{*}}{r}\right)^{-1} d r^{2}-r^{2}\left[d \theta^{2}+\sin (\theta)^{2} d \varphi^{2}\right], \tag{2.74}
\end{equation*}
where r_(**)=2GMr_{*}=2 G M and c=1c=1, can be embedded in 5+1-dimensional Minkowski spacetime with coordinates (Z_(1),Z_(2),dots,Z_(6))\left(Z_{1}, Z_{2}, \ldots, Z_{6}\right) and metric
defines a curved metric on M_(3)M_{3}. Determine the lightlike geodesics with constant spherical angle phi\phi (where x^(1)=r sin theta cos phi,x^(2)=r sin theta sin phix^{1}=r \sin \theta \cos \phi, x^{2}=r \sin \theta \sin \phi, and x^(3)=r cos thetax^{3}=r \cos \theta as usual).
Problem 2.79 The Minkowski metric ds^(2)=(dx^(0))^(2)-(dx^(1))^(2)-(dx^(2))^(2)d s^{2}=\left(d x^{0}\right)^{2}-\left(d x^{1}\right)^{2}-\left(d x^{2}\right)^{2} in R^(3)\mathbb{R}^{3} induces a nonflat Lorentzian metric on the surface S={(x^(0),x^(1),x^(2)):(x^(0))^(2)-(x^(1))^(2)-:}S=\left\{\left(x^{0}, x^{1}, x^{2}\right):\left(x^{0}\right)^{2}-\left(x^{1}\right)^{2}-\right.{:(x^(2))^(2)=-1}\left.\left(x^{2}\right)^{2}=-1\right\}. Let phi\phi be the polar angle in the (x^(1),x^(2))\left(x^{1}, x^{2}\right)-plane. Compute the global time difference Deltax^(0)\Delta x^{0} needed for a light signal to travel from a point phi_(0)=0\phi_{0}=0 to a point phi=pi//2\phi=\pi / 2 on SS.
Problem 2.80 Let (x^(0)(s),r(s),theta(s),phi(s))\left(x^{0}(s), r(s), \theta(s), \phi(s)\right) be a lightlike geodesic for the Schwarzschild metric, expressed in the spherical coordinates (r,theta,phi)(r, \theta, \phi). Derive a differential equation for r(s)r(s) in the form
when restricted to the plane theta=(pi)/(2)\theta=\frac{\pi}{2}.
Hint: The following nonzero Christoffel symbols for the Schwarzschild metric when theta=(pi)/(2)\theta=\frac{\pi}{2} might be useful: Gamma_(0r)^(0)=-Gamma_(rr)^(r)=(1)/(2alpha)(d alpha)/(dr),Gamma_(00)^(r)=(alpha)/(2)(d alpha)/(dr),Gamma_(theta theta)^(r)=Gamma_(phi phi)^(r)=-r alpha\Gamma_{0 r}^{0}=-\Gamma_{r r}^{r}=\frac{1}{2 \alpha} \frac{d \alpha}{d r}, \Gamma_{00}^{r}=\frac{\alpha}{2} \frac{d \alpha}{d r}, \Gamma_{\theta \theta}^{r}=\Gamma_{\phi \phi}^{r}=-r \alpha, and Gamma_(r theta)^(theta)=Gamma_(r phi)^(phi)=(1)/(r)\Gamma_{r \theta}^{\theta}=\Gamma_{r \phi}^{\phi}=\frac{1}{r}, where alpha=alpha(r)=1-(2GM)/(c^(2)r)\alpha=\alpha(r)=1-\frac{2 G M}{c^{2} r}.
Problem 2.81 Consider the metric ds^(2)=c^(2)dt^(2)-S(t)^(2)(dx^(2)+dy^(2)+dz^(2))d s^{2}=c^{2} d t^{2}-S(t)^{2}\left(d x^{2}+d y^{2}+d z^{2}\right), where S(t)S(t) is an increasing function of time tt with S(0)=0S(0)=0. Find the geodesic equations of motion. In particular, construct explicitly the lightlike geodesic when S(t)=t//t_(0)S(t)=t / t_{0} for some constant t_(0) > 0t_{0}>0. What are the points (ct,x,y,z)inR^(4)(c t, x, y, z) \in \mathbb{R}^{4} for a fixed t > t_(0)t>t_{0}, which are causally related to the event p=(ct_(0),ct_(0),0,0)p=\left(c t_{0}, c t_{0}, 0,0\right), i.e., the points which are connected to pp by a future-directed timelike (or lightlike) curve?
Problem 2.82 The line element on the unit sphere S^(2)\mathbb{S}^{2} is given by
{:(2.78)ds^(2)=dtheta^(2)+sin^(2)(theta)dvarphi^(2)",":}\begin{equation*}
d s^{2}=d \theta^{2}+\sin ^{2}(\theta) d \varphi^{2}, \tag{2.78}
\end{equation*}
Consider two geodesics separated by a small distance delta\delta and both orthogonal to the equator theta=pi//2\theta=\pi / 2, compute the rate of acceleration of the geodesics toward each other through the geodesic deviation equation
{:(2.80)A^(a)=R_(bcd)^(a)chi^(˙)^(b)chi^(˙)^(c)X^(d)",":}\begin{equation*}
A^{a}=R_{b c d}^{a} \dot{\chi}^{b} \dot{\chi}^{c} X^{d}, \tag{2.80}
\end{equation*}
where X^(d)X^{d} is the infinitesimal separation of the geodesics at the equator and chi^(a)\chi^{a} are the coordinates.
Problem 2.83 The metric for the de Sitter universe can be expressed in the form
{:(2.81)ds^(2)=dt^(2)-e^(2t//R)(dx^(2)+dy^(2)+dz^(2))",":}\begin{equation*}
d s^{2}=d t^{2}-e^{2 t / R}\left(d x^{2}+d y^{2}+d z^{2}\right), \tag{2.81}
\end{equation*}
where R > 0R>0 is a constant and x,yx, y, and zz can be treated as rectangular coordinates and tt as time.
a) Show that the trajectories of freely falling particles and photons are straight lines.
b) A body at a point x=X > 0x=X>0 on the xx-axis emits a photon toward the origin at time t=0t=0. Show that, if X < RX<R, the photon arrives at the origin at t=-R log(1-X//R)t=-R \log (1-X / R).
Problem 2.84 The de Sitter universe dS_(4)\mathrm{dS}_{4} is defined as the hyperboloid
in units with c=1c=1 and where T_(0) > 0T_{0}>0 is a constant. The metric on dS_(4)\mathrm{dS}_{4} is defined by restricting the five-dimensional Minkowski metric to the hyperboloid.
a) Find an explicit expression for the four-dimensional metric in a suitable coordinate system on dS_(4)\mathrm{dS}_{4}.
b) Derive the geodesic equations in dS_(4)\mathrm{dS}_{4}.
c) Compare the metric tensor with the Robertson-Walker metric
{:(2.83)ds^(2)=dt^(2)-S(t)^(2)dOmega^(2)",":}\begin{equation*}
d s^{2}=d t^{2}-S(t)^{2} d \Omega^{2}, \tag{2.83}
\end{equation*}
by writing down your metric in a coordinate system where the coefficient in front of the timelike coordinate is identically equal to one.
Problem 2.85 Consider AdS_(2)\mathrm{AdS}_{2} and the coordinates x^(mu)x^{\mu} defined in Problem 2.27.
a) Find the trajectory of a light ray in this spacetime.
b) A particle at rest in r=r_(0) > 2r=r_{0}>2 starts to fall freely at t=0t=0. What is the proper time it takes for the particle to freely fall and reach r=r_(1)r=r_{1}, where 1 < r_(1) < r_(0)1<r_{1}<r_{0} ? Also compute the coordinate time for this fall, i.e., the time tt at which the particle reaches r_(1)r_{1}. Discuss your result in the limit r_(1)rarr1r_{1} \rightarrow 1, and in particular, a possible physical interpretation of what you find. (Your answers may contain integrals and functions defined by implicit equations.)
Problem 2.86 Four-dimensional anti-de Sitter space, which is also called AdS_(4)\mathrm{AdS}_{4}, is a four-dimensional curved space that can be defined as follows: Consider the fivedimensional space with coordinates (X^(a))=(U,V,X,Y,Z)\left(X^{a}\right)=(U, V, X, Y, Z), where a=1,2,3,4,5a=1,2,3,4,5, and metric
{:(2.84)ds^(2)=dU^(2)+dV^(2)-dX^(2)-dY^(2)-dZ^(2)-=dX^(a)dX_(a):}\begin{equation*}
d s^{2}=d U^{2}+d V^{2}-d X^{2}-d Y^{2}-d Z^{2} \equiv d X^{a} d X_{a} \tag{2.84}
\end{equation*}
Then, AdS_(4)\mathrm{AdS}_{4} is the subspace of this space defined by the relation
Hint: Note that (X_(a))=(U,V,-X,-Y,-Z)\left(X_{a}\right)=(U, V,-X,-Y,-Z).
a) Compute the metric of AdS_(4)\operatorname{AdS}_{4} in the coordinate system (x^(mu))=(alpha,lambda,theta,varphi)\left(x^{\mu}\right)=(\alpha, \lambda, \theta, \varphi), where mu=0,1,2,3\mu=0,1,2,3, defined as follows: Let (r,theta,varphi)(r, \theta, \varphi) be spherical coordinates for (X,Y,Z)(X, Y, Z), i.e., X=r sin(theta)cos(varphi),Y=r sin(theta)sin(varphi)X=r \sin (\theta) \cos (\varphi), Y=r \sin (\theta) \sin (\varphi), and Z=r cos(theta)Z=r \cos (\theta), and (t,alpha)(t, \alpha) polar coordinates for (U,V)(U, V), i.e., U=t cos(alpha)U=t \cos (\alpha) and V=t sin(alpha)V=t \sin (\alpha). Then, t=cosh(lambda)t=\cosh (\lambda) and r=sinh(lambda)r=\sinh (\lambda).
b) Compute all possible trajectories lambda(alpha)\lambda(\alpha) of light rays on AdS_(4)\mathrm{AdS}_{4} moving along the subspace Y=Z=0Y=Z=0.
c) Prove that all lightlike geodesics on AdS_(4)\mathrm{AdS}_{4} are straight lines in the embedding space, i.e., they obey the equations
with the dot indicating differentiation with respect to proper time tau\tau.
Hint: You can find these trajectories by extremizing the functional intLd tau\int \mathcal{L} d \tau with
and lambda\lambda a Lagrange multiplier (explain why this is so!). One key step in the proof is to show that X^(˙)^(a)X^(˙)_(a)\dot{X}^{a} \dot{X}_{a} is conserved and you can assume that it is identically equal to zero for a lightlike trajectory.
Problem 2.87 Consider the 1+1-dimensional Robertson-Walker spacetime described by the metric
{:(2.88)ds^(2)=dt^(2)-a(t)^(2)dx^(2):}\begin{equation*}
d s^{2}=d t^{2}-a(t)^{2} d x^{2} \tag{2.88}
\end{equation*}
for some function a(t)a(t).
a) Compute the Ricci tensor R_(mu nu)R_{\mu \nu} for this spacetime.
b) Derive the trajectory x(t)x(t) of a light ray on this spacetime for a(t)=1//(A^(2)+:}a(t)=1 /\left(A^{2}+\right.B^(2)t^(2)B^{2} t^{2} ) and some constants AA and BB. Assume that the light ray is emitted at t=0t=0 from x=x_(0)x=x_{0} with dx//dt > 0d x / d t>0.
Problem 2.88 Consider the 1+2-dimensional Robertson-Walker spacetime described by the metric
{:(2.89)ds^(2)=dt^(2)-a(t)^(2)((dr^(2))/(1-kr^(2))+r^(2)dphi^(2)):}\begin{equation*}
d s^{2}=d t^{2}-a(t)^{2}\left(\frac{d r^{2}}{1-k r^{2}}+r^{2} d \phi^{2}\right) \tag{2.89}
\end{equation*}
where a(t)a(t) is some function and kk is a constant. Derive the geodesic equations and determine the Christoffel symbols.
Problem 2.89 The Robertson-Walker metric describing a particular closed universe is given by
{:(2.90)ds^(2)=dt^(2)-e^(-2t//a)[(1)/(1+(r//a)^(2))dr^(2)+r^(2)(sin^(2)theta dvarphi^(2)+dtheta^(2))]",":}\begin{equation*}
d s^{2}=d t^{2}-e^{-2 t / a}\left[\frac{1}{1+(r / a)^{2}} d r^{2}+r^{2}\left(\sin ^{2} \theta d \varphi^{2}+d \theta^{2}\right)\right], \tag{2.90}
\end{equation*}
for some parameter a > 0a>0.
a) Determine all nonzero Christoffel symbols Gamma_(mu nu)^(r),mu,nu=t,r,theta,varphi\Gamma_{\mu \nu}^{r}, \mu, \nu=t, r, \theta, \varphi for this metric. Moreover, for the vector field with the components A^(t)=t//a,A^(r)=r//a,A^(theta)=A^{t}=t / a, A^{r}=r / a, A^{\theta}=A^(varphi)=0A^{\varphi}=0 in the coordinate vector basis, compute the components grad_(mu)A^(r),mu=t,r\nabla_{\mu} A^{r}, \mu=t, r, of the covariant derivative.
b) Find the trajectory r(t)r(t) of a light pulse emitted at time t=0t=0 at r=ar=a and moving radially outward such that theta=pi//2\theta=\pi / 2 and varphi=0\varphi=0 at all times.
Problem 2.90 The Robertson-Walker metric is defined by
{:(2.91)ds^(2)=c^(2)dt^(2)-S(t)^(2)((dr^(2))/(1-kr^(2))+r^(2)dOmega^(2)):}\begin{equation*}
d s^{2}=c^{2} d t^{2}-S(t)^{2}\left(\frac{d r^{2}}{1-k r^{2}}+r^{2} d \Omega^{2}\right) \tag{2.91}
\end{equation*}
for some smooth function S(t)S(t) and dOmega^(2)=dtheta^(2)+sin^(2)theta dphi^(2)d \Omega^{2}=d \theta^{2}+\sin ^{2} \theta d \phi^{2}. We consider the case k=1k=1. After a coordinate transformation chi=arcsin r\chi=\arcsin r (with 0 <= chi <= pi//20 \leq \chi \leq \pi / 2 ), this can be written as
{:(2.92)ds^(2)=c^(2)dt^(2)-S(t)^(2)(dchi^(2)+sin^(2)chi dOmega^(2)):}\begin{equation*}
d s^{2}=c^{2} d t^{2}-S(t)^{2}\left(d \chi^{2}+\sin ^{2} \chi d \Omega^{2}\right) \tag{2.92}
\end{equation*}
a) Derive first integrals for the geodesic equations when d Omega=0d \Omega=0.
b) Derive a formula expressing the distance a light ray emitted at r=0r=0 at universal time t_(0)t_{0} travels (in the rr coordinate) in the time interval [t_(0),t_(0)+T]\left[t_{0}, t_{0}+T\right].
Problem 2.91 A spaceship is freely falling (along a geodesic) toward the true singularity at r=0r=0 in a Schwarzschild black hole. The initial velocity is theta^(˙)=phi^(˙)=0\dot{\theta}=\dot{\phi}=0, r^(˙)=alpha\dot{r}=\alpha, and t^(˙)=beta\dot{t}=\beta, where the dot means differentiation with respect to the path
parameter (which can be taken to be the proper time) and the standard Schwarzschild metric is used, i.e.,
{:(2.93)ds^(2)=(1-(2GM)/(c^(2)r))c^(2)dt^(2)-(1-(2GM)/(c^(2)r))^(-1)dr^(2)-r^(2)dOmega^(2):}\begin{equation*}
d s^{2}=\left(1-\frac{2 G M}{c^{2} r}\right) c^{2} d t^{2}-\left(1-\frac{2 G M}{c^{2} r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2} \tag{2.93}
\end{equation*}
The proper time tau\tau needed to reach the singularity r=0r=0, when starting from r=r=r_(0) < 2GM//c^(2)r_{0}<2 G M / c^{2}, can be written as
{:(2.94)tau=int_(0)^(r_(0))f(r)dr:}\begin{equation*}
\tau=\int_{0}^{r_{0}} f(r) d r \tag{2.94}
\end{equation*}
What is the function f(r)f(r) ?
Problem 2.92 A Schwarzschild black hole has a mass M=13.5*10^(30)kgM=13.5 \cdot 10^{30} \mathrm{~kg} (about seven times the solar mass). An observer is freely falling (along a geodesic) radially toward the black hole. The initial radial coordinate is r=r_(0)=10^(10)kmr=r_{0}=10^{10} \mathrm{~km} and the initial coordinate velocity is c(dr//dx^(0))=-v_(0)=-10km//sc\left(d r / d x^{0}\right)=-v_{0}=-10 \mathrm{~km} / \mathrm{s}. Derive the formula for the proper time needed to reach the Schwarzschild horizon and give the order of magnitude of this time. Newton's gravitational constant is G~~6.67*10^(-11)m^(3)//(kg*s^(2))G \approx 6.67 \cdot 10^{-11} \mathrm{~m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right) and the speed of light is c~~3*10^(8)m//sc \approx 3 \cdot 10^{8} \mathrm{~m} / \mathrm{s}.
Hint: The following integral can be useful int(dx)/(sqrt(a+(b)/(x)))=(1)/(sqrta)[sqrt(x^(2)+(bx)/(a))-(b)/(2a)ln(x+(b)/(2a)+sqrt(x^(2)+(bx)/(a)))]+C\int \frac{d x}{\sqrt{a+\frac{b}{x}}}=\frac{1}{\sqrt{a}}\left[\sqrt{x^{2}+\frac{b x}{a}}-\frac{b}{2 a} \ln \left(x+\frac{b}{2 a}+\sqrt{x^{2}+\frac{b x}{a}}\right)\right]+C,
where a,ba, b, and CC are constants.
Problem 2.93 A particle of mass m > 0m>0 is freely falling radially toward the horizon of a Schwarzschild black hole of mass MM. Show that p_(0)=mcg_(00)x^(˙)^(0)p_{0}=m c g_{00} \dot{x}^{0} is a constant of motion. Find the proper time Delta s\Delta s (as a function of p_(0)=E//cp_{0}=E / c ) needed for the particle to reach r=2GM//c^(2)r=2 G M / c^{2} from r=3GM//c^(2)r=3 G M / c^{2}. Show that the result can be written as
where r_(**)-=2GM//c^(2)r_{*} \equiv 2 G M / c^{2}.
Problem 2.94 An observer in the Schwarzschild spacetime moves with fixed radial coordinate r=r_(0)r=r_{0} and fixed angular velocity varphi^(˙)=omega\dot{\varphi}=\omega in the plane theta=pi//2\theta=\pi / 2. Compute the 4 -acceleration AA and the proper acceleration alpha\alpha of the observer as a function of the proper period omega\omega.
Problem 2.95 A particle of mass m > 0m>0 is freely falling radially toward the event horizon r=2GMr=2 G M of a Schwarzschild black hole of mass MM (we set c=1c=1 ), i.e., theta\theta and varphi\varphi are constant in the standard coordinates where the metric is ds^(2)=(1-(2GM)/(r))dt^(2)-(1-(2GM)/(r))^(-1)dr^(2)-r^(2)[dtheta^(2)+sin(theta)^(2)dvarphi^(2)]d s^{2}=\left(1-\frac{2 G M}{r}\right) d t^{2}-\left(1-\frac{2 G M}{r}\right)^{-1} d r^{2}-r^{2}\left[d \theta^{2}+\sin (\theta)^{2} d \varphi^{2}\right].
a) Compute the trajectory of this particle.
b) Find also formulas for the coordinate time Delta t\Delta t and proper time Delta tau\Delta \tau it takes for the particle to reach the event horizon from some position r=r_(0)r=r_{0}. Determine for each of these times if it is finite or infinite, and discuss the physical significance of your results.
Integration constants can be fixed so that the results have a simple form, but if so a physical interpretation of what the choice amounts to should be given.
Problem 2.96 Consider an observer in the Schwarzschild spacetime with line element
{:(2.98)ds^(2)=(1-(r_(**))/(r))dt^(2)-(1-(r_(**))/(r))^(-1)dr^(2)-r^(2)dOmega^(2):}\begin{equation*}
d s^{2}=\left(1-\frac{r_{*}}{r}\right) d t^{2}-\left(1-\frac{r_{*}}{r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2} \tag{2.98}
\end{equation*}
The observer starts out at r=r_(0)r=r_{0} and initially moves tangentially with a local velocity v_(0)v_{0} relative to the stationary frame.
a) Determine the minimal value of v_(0)v_{0} such that the observer does not fall into the black hole region of the solution.
b) Assuming that v_(0)=0v_{0}=0, compute the proper time it takes for the observer to reach the singularity.
Hint: The local velocity vv of an observer relative to the stationary frame has a gamma\gamma factor of gamma=V*U\gamma=V \cdot U, where VV is the 4-velocity of the observer, U=alphadel_(t)U=\alpha \partial_{t}, and U^(2)=1U^{2}=1.
Problem 2.97 You are sending up a satellite around the Earth. You want it to be directed such that when you turn off its engines, it will follow a geodesic around the Earth with fixed radius. The metric around the Earth is ds^(2)=(1-(r_(**))/(r))dt^(2)-(1-(r_(**))/(r))^(-1)dr^(2)-r^(2)dtheta^(2)-r^(2)sin^(2)theta dphi^(2),quadR_(0) > r_(**)d s^{2}=\left(1-\frac{r_{*}}{r}\right) d t^{2}-\left(1-\frac{r_{*}}{r}\right)^{-1} d r^{2}-r^{2} d \theta^{2}-r^{2} \sin ^{2} \theta d \phi^{2}, \quad R_{0}>r_{*},
where R_(0)R_{0} is the radius of the Earth. Your initial data when you turn off the engines at tau=0\tau=0 are the following
Is this possible? If so, determine how your initial condition BB, which you have to choose, depends on RR and r_(**)r_{*}.
Problem 2.98 A satellite moves at a constant radial distance from the Earth with a constant orbital coordinate speed v=rd phi//dtv=r d \phi / d t. Assume that the metric is the Schwarzschild metric and let the orbit be in the plane with angle theta=pi//2\theta=\pi / 2 such that
{:(2.101)ds^(2)=(1-(r_(**))/(r))dt^(2)-(1-(r_(**))/(r))^(-1)dr^(2)-r^(2)dphi^(2):}\begin{equation*}
d s^{2}=\left(1-\frac{r_{*}}{r}\right) d t^{2}-\left(1-\frac{r_{*}}{r}\right)^{-1} d r^{2}-r^{2} d \phi^{2} \tag{2.101}
\end{equation*}
where r_(**)-=2GMr_{*} \equiv 2 G M is the Schwarzschild radius.
a) Calculate the proper time tau\tau for the satellite to complete one orbit around the Earth. Express the answer in terms of the coordinate speed vv and the radius rr.
b) Use the result in a) to calculate t//taut / \tau and show that if this is series expanded to first order in vv and the gravitational potential, it holds that
where Phi_(s)\Phi_{s} is the gravitational potential at the satellite.
Problem 2.99 The spacetime outside of the Earth may be approximately described by the Schwarzschild line element
{:(2.103)ds^(2)=(1-(r_(**))/(r))dt^(2)-(1-(r_(**))/(r))^(-1)dr^(2)-r^(2)dOmega^(2):}\begin{equation*}
d s^{2}=\left(1-\frac{r_{*}}{r}\right) d t^{2}-\left(1-\frac{r_{*}}{r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2} \tag{2.103}
\end{equation*}
where r_(**)r_{*} is the Schwarzschild radius of the Earth (approximately 9 mm ). A GPS satellite is orbiting the Earth in free fall at a stationary radius r=r_(0)r=r_{0}. The motion is assumed to occur in the plane theta=pi//2\theta=\pi / 2.
a) Since rr is constant, the motion will have a 4 -velocity U=alphadel_(t)+betadel_(varphi)U=\alpha \partial_{t}+\beta \partial_{\varphi}. Find the values of the constants alpha\alpha and beta\beta.
b) Find an expression for the proper time it takes for the satellite to complete a full orbit around the Earth.
c) An observer is stationary at r=r_(0)r=r_{0} (note that this requires proper acceleration of this observer). At what speed will the satellite pass by the observer?
Hint: The relative speed vv between two objects with 4 -velocities UU and VV, respectively, has a gamma\gamma factor of gamma=V*U\gamma=V \cdot U.
Problem 2.100 Consider two observers in the exterior Schwarzschild spacetime with line element
{:(2.104)ds^(2)=(1-(r_(**))/(r))dt^(2)-(1-(r_(**))/(r))^(-1)dr^(2)-r^(2)dOmega^(2):}\begin{equation*}
d s^{2}=\left(1-\frac{r_{*}}{r}\right) d t^{2}-\left(1-\frac{r_{*}}{r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2} \tag{2.104}
\end{equation*}
Both observers can be assumed to move in the plane theta=pi//2\theta=\pi / 2. The first observer is a stationary observer with fixed spatial coordinates r=r_(0) > 3r_(**)r=r_{0}>3 r_{*} and varphi=varphi_(0)\varphi=\varphi_{0}, whereas the second observer is moving on a circular geodesic with radius r=r_(0)r=r_{0}.
a) Give a parametrization of the worldline for each observer and use it to find the proper acceleration of the observers.
b) The observers meet and synchronize their clocks when they pass each other. Find the ratio between the times shown by the respective clocks when they pass each other the next time.
c) Find the relative velocity of the observers as they pass each other.
Problem 2.101 You have reached a fast rotating neutron star with your spaceship. You decide that you want to go around the neutron star once. Let your orbit be at constant radial distance and your coordinate speed v=rd phi//dtv=r d \phi / d t.
a) Calculate the proper time tau\tau it takes you to go around. Express your answer in terms of the radius RR and the speed vv (set c=1c=1 ). The metric describing this neutron star is given by the Kerr metric, i.e.,
{:[ds^(2)=(1-(rr_(**))/(rho^(2)))dt^(2)+(2arr_(**)sin^(2)theta)/(rho^(2))dtd phi-(rho^(2))/(Delta)dr^(2)-rho^(2)dtheta^(2)],[(2.105)-(r^(2)+a^(2)+(a^(2)rr_(**)sin^(2)theta)/(rho^(2)))sin^(2)theta dphi^(2)]:}\begin{align*}
d s^{2}= & \left(1-\frac{r r_{*}}{\rho^{2}}\right) d t^{2}+\frac{2 a r r_{*} \sin ^{2} \theta}{\rho^{2}} d t d \phi-\frac{\rho^{2}}{\Delta} d r^{2}-\rho^{2} d \theta^{2} \\
& -\left(r^{2}+a^{2}+\frac{a^{2} r r_{*} \sin ^{2} \theta}{\rho^{2}}\right) \sin ^{2} \theta d \phi^{2} \tag{2.105}
\end{align*}
where rho^(2)-=r^(2)+a^(2)cos^(2)theta,Delta-=r^(2)-rr_(**)+a^(2),a-=J//M\rho^{2} \equiv r^{2}+a^{2} \cos ^{2} \theta, \Delta \equiv r^{2}-r r_{*}+a^{2}, a \equiv J / M, and MM and JJ are the mass and the angular momentum of the neutron star, respectively. You choose to make the orbit at a fixed angle of theta=pi//2\theta=\pi / 2.
b) Now, use your result in a) to calculate T//tauT / \tau, where TT is the coordinate time of your orbit, and show if you expand to first approximation in vv and the gravitational potential, you obtain
{:(2.106)(T)/( tau)≃1+(v^(2))/(2)[1+((a)/(R))^(2)]+(r_(**))/(2R)","quad" if "(r_(**))/(R)∼v^(2):}\begin{equation*}
\frac{T}{\tau} \simeq 1+\frac{v^{2}}{2}\left[1+\left(\frac{a}{R}\right)^{2}\right]+\frac{r_{*}}{2 R}, \quad \text { if } \frac{r_{*}}{R} \sim v^{2} \tag{2.106}
\end{equation*}
Problem 2.102 At the time of inflation, consider a massive free-falling particle. Assume that the metric of spacetime is
{:(2.107)ds^(2)=dt^(2)-a(t)^(2)[drho^(2)+rho^(2)(dtheta^(2)+sin^(2)theta dphi^(2))]","quad a(t)=a_(0)e^(Ct)",":}\begin{equation*}
d s^{2}=d t^{2}-a(t)^{2}\left[d \rho^{2}+\rho^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right], \quad a(t)=a_{0} e^{C t}, \tag{2.107}
\end{equation*}
where a_(0)a_{0} and CC are constants. The initial values for the free-falling particle are given by
a) Is the spatial geometry curved or not (at each fixed value of time)? Justify your answer.
b) Calculate the proper time for the free-falling particle between the coordinate times t=0t=0 and t=t_(1)t=t_{1}.
Hint: Calculations might become simpler if another coordinate system is used! It is also okay to give the answer expressed as an integral.
Problem 2.103 Assume a three-dimensional version of the Robertson-Walker metric (with k=0k=0 ):
{:(2.109)ds^(2)=dt^(2)-a(t)^(2)(dr^(2)+r^(2)dphi^(2)):}\begin{equation*}
d s^{2}=d t^{2}-a(t)^{2}\left(d r^{2}+r^{2} d \phi^{2}\right) \tag{2.109}
\end{equation*}
You are traveling in your spaceship in this universe. You decide to travel in a circle around r=0r=0 on a fixed radius r=R_(0)r=R_{0}.
a) Calculate the proper time for the spaceship in this geometry going around one time (let phi\phi go from zero to 2pi2 \pi ) if you have the following constant velocity v=a(t)R_(0)(d phi)/(dt)v=a(t) R_{0} \frac{d \phi}{d t} and a(t)=exp(t)a(t)=\exp (t). Let the initial time be t=0t=0.
b) Is it always possible to get around in a finite time?
Problem 2.104 You are traveling in your spaceship in outer intergalactic space. The metric can be assumed to be the Robertson-Walker metric with zero curvature:
{:(2.110)ds^(2)=dt^(2)-a(t)^(2)(dx^(2)+dy^(2)+dz^(2)):}\begin{equation*}
d s^{2}=d t^{2}-a(t)^{2}\left(d x^{2}+d y^{2}+d z^{2}\right) \tag{2.110}
\end{equation*}
To save fuel, you do not use the spaceship's engines. You are moving according to the following initial conditions
Calculate the proper time it takes you to reach x=X_(D)x=X_{D}. It is sufficient to give your answer in terms of an integral, which depends on the initial value AA and the scale factor a(t)a(t).
Problem 2.105 An observer (A)(A) is stationary in Schwarzschild spacetime at a radius r_(0)r_{0}. A second observer (B)(B) is initially stationary at r=r_(1)r=r_{1}, but at some event (which can be assigned t=tau=0t=\tau=0 ) suffers from a rocket failure and becomes freely falling. On the way into the black hole, BB passes right by AA.
a) What is the relative velocity of AA and BB as they pass by each other?
b) What proper time has passed for BB since the rocket failure when they pass by each other?
Problem 2.106 For the two-dimensional spacetime with coordinates tt and xx and line element ds^(2)=x^(2)dt^(2)-dx^(2)d s^{2}=x^{2} d t^{2}-d x^{2} :
a) Compute the proper acceleration of an observer with worldline x=x_(0)x=x_{0}, where x_(0)x_{0} is a constant.
b) Compute the proper time for a free-falling observer starting at x=x_(0)x=x_{0} with dx//dt=0d x / d t=0 to reach the coordinate singularity at x=0x=0.
2.7 Kruskal-Szekeres Coordinates
Problem 2.107 Show that for r > 2mur>2 \mu the Kruskal-Szekeres metric
{:(2.112)ds^(2)=(16mu^(2))/(r)e^((2mu-r)//2mu)dudv-r^(2)(dtheta^(2)+sin^(2)theta dphi^(2)):}\begin{equation*}
d s^{2}=\frac{16 \mu^{2}}{r} e^{(2 \mu-r) / 2 \mu} d u d v-r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) \tag{2.112}
\end{equation*}
is equivalent to the standard Schwarzschild metric through the relations
Here u < 0u<0 and v > 0v>0, and we use units with c=1c=1.
Problem 2.108 Show that a spaceship cannot get out from the black hole region u > 0u>0 and v > 0v>0 in Kruskal-Szekeres coordinates.
Problem 2.109 An observer is freely falling to the true singularity r=0r=0 of a Schwarzschild black hole. We assume that the fall follows a radial ray d Omega=0d \Omega=0. Since the standard (spherical) coordinates become singular at the Schwarzschild event horizon r=2GM//c^(2)-=2mur=2 G M / c^{2} \equiv 2 \mu, we express the initial condition of the observer in terms of the Kruskal-Szekeres coordinates
and v^(˙)(0)\dot{v}(0) is determined by the requirement of the 4 -velocity being future-directed with modulus one. Compute the proper time Delta s\Delta s for the fall, expressed as an integral
int_(0)^(2mu)f(r)dr\int_{0}^{2 \mu} f(r) d r
for a certain function f(r)f(r) of the radius rr expressed in terms of EE and v_(0)v_{0}.
Hint: From the equations of motion, it can be shown that the quantity
is a constant of motion.
Problem 2.110 Consider a Schwarzschild black hole with metric
{:(2.116)ds^(2)=(1-(r_(**))/(r))dt^(2)-(1-(r_(**))/(r))^(-1)dr^(2)-r^(2)dOmega^(2):}\begin{equation*}
d s^{2}=\left(1-\frac{r_{*}}{r}\right) d t^{2}-\left(1-\frac{r_{*}}{r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2} \tag{2.116}
\end{equation*}
a) What conclusion can you draw about the singularity r=0r=0 of this metric from the fact that
{:(2.117)R_(mu v alpha beta)R^(mu v alpha beta)=(3r_(**)^(2))/(r^(6))?:}\begin{equation*}
R_{\mu v \alpha \beta} R^{\mu v \alpha \beta}=\frac{3 r_{*}^{2}}{r^{6}} ? \tag{2.117}
\end{equation*}
Do not forget to motivate your answer.
b) Determine what radial light cones look like using the Schwarzschild metric, i.e., how tt depends on rr.
c) Instead of using the Schwarzschild coordinates to describe black holes, it can be useful to use Kruskal-Szekeres coordinates. The metric then takes the form
{:(2.118)ds^(2)=(4r_(**)^(3))/(r)*e^(-r//r_(**))(dV^(2)-dU^(2))-r^(2)dOmega^(2):}\begin{equation*}
d s^{2}=\frac{4 r_{*}^{3}}{r} \cdot e^{-r / r_{*}}\left(d V^{2}-d U^{2}\right)-r^{2} d \Omega^{2} \tag{2.118}
\end{equation*}
Determine what radial light cones look like in these coordinates.
2.8 Weak Field Approximation and Newtonian Limit
Problem 2.111 a) What are the equations of motion for a massive particle in a gravitational potential according to Newton's mechanics and general relativity, respectively? Derive the former from the latter in the Newtonian limit.
b) What are tidal forces in Newton's theory of gravity? How are they related to the gravitational potential? Why are solar tidal forces slightly weaker than lunar tidal forces? In general relativity, explain how the tidal forces are identified with the curvature of spacetime.
Problem 2.112 a) Compute the Ricci tensor R_(mu nu)R_{\mu \nu} in the linear approximation for a metric g_(mu nu)=eta_(mu nu)+h_(mu nu)g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}, i.e., you can ignore all but the first-order terms in h_(mu nu)h_{\mu \nu}.
b) Show that a coordinate transformation
the linearized Einstein equations R_(mu nu)=0R_{\mu \nu}=0 reduce to the wave equation for bar(h)_(mu nu)\bar{h}_{\mu \nu}.
Problem 2.113 The spacetime metric corresponding to a weak gravitational potential |Phi(x)|≪c^(2)|\Phi(\mathbf{x})| \ll c^{2} is
{:(2.122)ds^(2)=[c^(2)-2Phi(x)]dt^(2)-[1+(2Phi(x))/(c^(2))](dx^(2)+dy^(2)+dz^(2)):}\begin{equation*}
d s^{2}=\left[c^{2}-2 \Phi(\mathbf{x})\right] d t^{2}-\left[1+\frac{2 \Phi(\mathbf{x})}{c^{2}}\right]\left(d x^{2}+d y^{2}+d z^{2}\right) \tag{2.122}
\end{equation*}
a) Find the geodesic equation for this metric in the nonrelativistic and weak field (where you only keep the lowest-order terms in Phi\Phi ) limits. Discuss your result.
b) Compute the redshift of a photon with angular frequency omega\omega moving in the Earth's gravitational field Phi(x)=-gz\Phi(\mathbf{x})=-g z (independent of xx and yy ) from z=0z=0 to z=h > 0z=h>0 as observed by a stationary observer. You should give a detailed derivation of your result. Discuss also how one can understand the result using the equivalence principle.
Problem 2.114 Consider two massive particles moving freely on two close paths on a curved spacetime with metric ds^(2)=g_(mu nu)dx^(mu)dx^(nu)d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu} and assume that the positions of these two particles at proper time tau\tau are x^(mu)(tau)x^{\mu}(\tau) and x^(mu)(tau)+s^(mu)(tau)x^{\mu}(\tau)+s^{\mu}(\tau), respectively, with s^(mu)s^{\mu} small (i.e., only terms linear in s^(mu)s^{\mu} need to be taken into account and higherorder terms can be ignored).
a) Derive the geodesic deviation equation
b) Show that in the Newtonian limit the geodesic deviation equation reduces to the equation for tidal acceleration in Newton's theory of gravitation, i.e.,
To derive the equation in a) it is convenient, but not necessary, to work in a local inertial frame.
Problem 2.115 Derive a relativistic generalization of the centrifugal force as follows: Consider the motion of a free particle in Minkowski space in a coordinate system rotating with constant angular velocity omega\omega around the zz-coordinate axis.
a) Compute the Christoffel symbols and the geodesic equations in this coordinate system.
Hint: It is convenient to use cylindrical coordinates (r,varphi,z)(r, \varphi, z).
b) From your result in a), derive the equations of motion in this rotating frame in the nonrelativistic limit (for omega r≪1\omega r \ll 1 ).
Problem 2.116 a) Find the trajectory of a planet with mass mm moving on a circle in the gravitational potential V(r)=-GMm//|r|V(\mathbf{r})=-G M m /|\mathbf{r}|, according to Newton's mechanics.
b) There is a natural generalization of the trajectory in a) to general relativity. Explain what this generalization is. Find this generalized trajectory.
Hint: The trajectory can be computed from Hamilton's principle
{:(2.126)delta int((1)/(2)mr^(˙)^(2)+(GMm)/(r))dt=0:}\begin{equation*}
\delta \int\left(\frac{1}{2} m \dot{\mathbf{r}}^{2}+\frac{G M m}{r}\right) d t=0 \tag{2.126}
\end{equation*}
using spherical coordinates (r,theta,varphi)(r, \theta, \varphi) and assuming theta(t)=pi//2\theta(t)=\pi / 2 and r(t)=r_(0)=r(t)=r_{0}= constant. Recall that dx^(2)+dy^(2)+dz^(2)=dr^(2)+r^(2)(dtheta^(2)+sin^(2)theta dvarphi^(2))d x^{2}+d y^{2}+d z^{2}=d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \varphi^{2}\right). Find the relation between rr and the angular momentum L=mr^(2)varphi^(˙)L=m r^{2} \dot{\varphi}.
Problem 2.117 Consider the Einstein field equations.
a) What three approximations should be made to obtain the Newtonian limit?
b) Show that, in the Newtonian limit, the Einstein field equations reduce to
where Phi\Phi is the gravitational potential and rho\rho the mass density.
Problem 2.118 Consider a satellite in circular orbit around the Earth at a distance R_(1)R_{1} from the surface. The metric outside of the Earth can be considered to be
{:(2.128)ds^(2)=(1+2Phi)dt^(2)-(1+2Phi)^(-1)dr^(2)-r^(2)dOmega^(2):}\begin{equation*}
d s^{2}=(1+2 \Phi) d t^{2}-(1+2 \Phi)^{-1} d r^{2}-r^{2} d \Omega^{2} \tag{2.128}
\end{equation*}
where Phi=-GM//r\Phi=-G M / r is the classical gravitational potential and dOmega^(2)=dtheta^(2)+d \Omega^{2}=d \theta^{2}+sin^(2)theta dphi^(2)\sin ^{2} \theta d \phi^{2}. What is the eigentime required for the satellite to complete a full orbit around the Earth? How does this compare with the global time tt required for the same orbit?
2.9 Gravitational Lensing
Problem 2.119 Consider a spherical body with radius R_(0)R_{0}, constant density, and total mass M_(0)M_{0}. Neutrinos traveling through this body have such small masses that their worldlines can be roughly approximated as null geodesics. Find an expression for the angular deflection of a neutrino with an impact parameter (smallest distance to the body's center) b < R_(0)b<R_{0}. Verify that your expression has the expected limit when b rarrR_(0)b \rightarrow R_{0}. You may work in the low-velocity and weak field limits for computing the metric.
Problem 2.120 For large distances from the center of the halo, the Navarro-FrenkWhite (NFW) dark matter halo profile assumes that the matter density varies as rho(r)=k//r^(3)\rho(r)=k / r^{3}. Find the deflection angle alpha\alpha due to gravitational lensing of light that passes such a halo at a minimum distance r_(0)r_{0}. You may assume that the NFW density profile is valid from some radius r=r_(s) < r_(0)r=r_{s}<r_{0} and that the mass inside this radius is given by M_(0)M_{0}.
2.10 Frequency Shifts
Problem 2.121 a) Derive the formula for the gravitational redshift between stationary observers in a static spacetime with line element such that the metric components g_(mu nu)g_{\mu \nu} are independent of a global time coordinate tt.
b) Explain the origin of the gravitational redshift in the case of the Schwarzschild metric and derive the approximative formula
{:(2.129)z-=(lambda_(B)-lambda_(A))/(lambda_(A))≃(GM)/(c^(2)r_(A)):}\begin{equation*}
z \equiv \frac{\lambda_{B}-\lambda_{A}}{\lambda_{A}} \simeq \frac{G M}{c^{2} r_{A}} \tag{2.129}
\end{equation*}
for the redshift observed far away at BB, from a source at a radial coordinate r=r_(A)r=r_{A}.
Problem 2.122 A 1+1-dimensional universe is defined as the surface
{:(2.130)(ct)^(2)-x^(2)-y^(2)=-K^(2)quad(" where "K > 0):}\begin{equation*}
(c t)^{2}-x^{2}-y^{2}=-K^{2} \quad(\text { where } K>0) \tag{2.130}
\end{equation*}
in R^(3)\mathbb{R}^{3}. The metric on the surface is induced by the Minkowski metric ds^(2)=c^(2)dt^(2)-d s^{2}=c^{2} d t^{2}-dx^(2)-dy^(2)d x^{2}-d y^{2} in R^(3)\mathbb{R}^{3}. Analyze the frequency shift in this mini-universe between comoving observers.
Problem 2.123 A spaceship is moving radially toward a center of mass MM with a coordinate velocity dr//dt=-0.1 cd r / d t=-0.1 c, where tt is the Schwarzschild universal time and c≃3*10^(8)m//s^(2)c \simeq 3 \cdot 10^{8} \mathrm{~m} / \mathrm{s}^{2}. An observer in the spaceship is measuring the wavelength of a light signal from a distant star at rest. The light signal travels along the same radius as the observer. The wavelength at r rarr oor \rightarrow \infty is assumed to be 4000"Å"4000 \AA. What is the observed wavelength when GM=10^(20)m^(3)//s^(2)G M=10^{20} \mathrm{~m}^{3} / \mathrm{s}^{2} and r=10^(6)mr=10^{6} \mathrm{~m} ?
Problem 2.124 Compute the redshift of starlight emitted from the surface of a star with r_("star ")=7*10^(8)mr_{\text {star }}=7 \cdot 10^{8} \mathrm{~m} and mass M=2*10^(30)kgM=2 \cdot 10^{30} \mathrm{~kg}. Use the approximate values G~~6.67*10^(-11)m^(3)//(kg*s^(2))G \approx 6.67 \cdot 10^{-11} \mathrm{~m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right) and c~~3*10^(8)m//sc \approx 3 \cdot 10^{8} \mathrm{~m} / \mathrm{s}.
Problem 2.125 Elements in the chromosphere of the Sun emit sharp spectral lines. A student in relativity theory observes one such known spectral line in a spectrometer on Earth. According to general relativity, the emitted light is affected by the mass of the Sun. Calculate, using the general theory of relativity and to lowest order in the gravitational constant, the magnitude and sign of the relative frequency shift Delta v//v\Delta v / v of this spectral line. The solar mass is about 2.0*10^(30)kg2.0 \cdot 10^{30} \mathrm{~kg}, Newton's gravitational constant is G~~6.7*10^(-11)m^(3)//(kg*s^(2))G \approx 6.7 \cdot 10^{-11} \mathrm{~m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right), the speed of light is c~~3.0*10^(8)m//sc \approx 3.0 \cdot 10^{8} \mathrm{~m} / \mathrm{s}, the solar radius is about 7.8*10^(8)m7.8 \cdot 10^{8} \mathrm{~m}, and the average distance Sun-Earth is about 1.5*10^(11)m1.5 \cdot 10^{11} \mathrm{~m}.
Problem 2.126 A spaceship is launched from the ground station on Earth and is moving radially upward. When it is at an altitude of 1000 km , its velocity is only about 0.1km//s0.1 \mathrm{~km} / \mathrm{s}. At that moment, a light signal is sent from the spaceship and is observed at the ground station. Compute the red/blue shifts of the signal from the two most important physical effects. Newton's gravitational constant is G~~6.67G \approx 6.67. 10^(-11)m^(3)//(kg*s^(2))10^{-11} \mathrm{~m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right) and the radius and the mass of the Earth are R~~6.3*10^(3)kmR \approx 6.3 \cdot 10^{3} \mathrm{~km} and M~~5.98*10^(24)kgM \approx 5.98 \cdot 10^{24} \mathrm{~kg}, respectively.
Problem 2.127 Compute the blueshift of a light signal sent from a very distant spaceship and observed at the Earth. Assume that the spaceship is stationary in an approximately static spacetime. Useful information: The distance between the Sun and the Earth is approximately 1.5*10^(11)m1.5 \cdot 10^{11} \mathrm{~m}, the speed of light is c≃3*10^(8)m//sc \simeq 3 \cdot 10^{8} \mathrm{~m} / \mathrm{s}, GM_(o.)≃1.3*10^(20)m^(3)//s^(2)G M_{\odot} \simeq 1.3 \cdot 10^{20} \mathrm{~m}^{3} / \mathrm{s}^{2} and the gravitational potential of the Earth on its surface (normalized to zero at infinity) is -6.24*10^(7)m^(2)//s^(2)-6.24 \cdot 10^{7} \mathrm{~m}^{2} / \mathrm{s}^{2}.
Problem 2.128 A free-falling observer is moving radially away from a black hole with a local velocity that is just large enough to escape the gravitational pull. The free-falling observer is emitting light signals radially toward an observer stationary at infinity. Compute the frequency of the light signal received by the second observer if it was emitted with frequency f_(0)f_{0} at radius rr by the first observer.
Problem 2.129 In the two-dimensional spacetime introduced in Problem 2.106, a series of light signals is emitted from a free-falling observer starting at x=x_(0)x=x_{0} and received by an observer with a worldline x=x_(1)x=x_{1}, where x_(1)x_{1} is a constant. Find the redshift zz of the light signals as a function of the position x=x_(e)x=x_{e} where the signal was emitted by the free-falling observer.
Problem 2.130 Consider the Robertson-Walker metric written as
{:(2.131)ds^(2)=c^(2)dt^(2)-S(t)^(2)[(dr^(2))/(1-kr^(2))+r^(2)dOmega^(2)]:}\begin{equation*}
d s^{2}=c^{2} d t^{2}-S(t)^{2}\left[\frac{d r^{2}}{1-k r^{2}}+r^{2} d \Omega^{2}\right] \tag{2.131}
\end{equation*}
for some fixed parameter k > 0k>0. We project the metric to two dimensions by setting d Omega=0d \Omega=0. An observer AA, located at ( t_(0),r_(0)t_{0}, r_{0} ) and at rest with respect to the coordinate rr, sends a light signal. Another observer, located at (t_(1),r_(1))\left(t_{1}, r_{1}\right) and also at rest with respect to rr, receives the light signal. After a short time epsilon,A\epsilon, A sends another light signal, which is received by BB at the time t_(1)+epsilon^(')t_{1}+\epsilon^{\prime}. Compute the ratio epsilon^(')//epsilon\epsilon^{\prime} / \epsilon in terms of the unknown function S(t)S(t) and deduce from this the cosmological redshift.
Problem 2.131 The Robertson-Walker metric (for k=1k=1 ) can be written as
{:(2.132)ds^(2)=c^(2)dt^(2)-S(t)^(2)[dchi^(2)+sin^(2)chi(dtheta^(2)+sin^(2)theta dphi^(2))]:}\begin{equation*}
d s^{2}=c^{2} d t^{2}-S(t)^{2}\left[d \chi^{2}+\sin ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right] \tag{2.132}
\end{equation*}
where 0 <= chi <= pi//2,0 <= theta <= pi0 \leq \chi \leq \pi / 2,0 \leq \theta \leq \pi, and 0 <= phi < 2pi0 \leq \phi<2 \pi. Derive the differential equations for the geodesics.
2.11 Gravitational Waves
Problem 2.132 Consider the weak field limit in the harmonic gauge, where g_(ab)=g_{a b}=eta_(ab)+epsih_(ab)\eta_{a b}+\varepsilon h_{a b} and g^(ab)grad_(a)grad_(b)chi^(c)=0g^{a b} \nabla_{a} \nabla_{b} \chi^{c}=0 for the coordinates chi^(a)\chi^{a}.
a) Show that the harmonic gauge leaves a residual gauge freedom chi^(a)|->\chi^{a} \mapstochi^('a)=chi^(a)+epsixi^(a)\chi^{\prime a}=\chi^{a}+\varepsilon \xi^{a}, which also preserves the weak field approximation, as long as g^(ab)grad_(a)grad_(b)xi^(c)=0g^{a b} \nabla_{a} \nabla_{b} \xi^{c}=0 and find an expression for h_(ab)^(')h_{a b}^{\prime} in the coordinates chi^('a)\chi^{\prime a}.
b) Defining bar(h)_(ab)=h_(ab)-g_(ab)h//2\bar{h}_{a b}=h_{a b}-g_{a b} h / 2, where h=h_(a)^(a)h=h_{a}^{a}, show that
in the harmonic gauge to leading order in epsi\varepsilon.
c) A gravitational plane wave satisfies bar(h)_(ab)=A_(ab)exp(ik_(c)chi^(c))\bar{h}_{a b}=A_{a b} \exp \left(i k_{c} \chi^{c}\right), where the wave equation can be used to conclude that eta_(ab)k^(a)k^(b)=0\eta_{a b} k^{a} k^{b}=0. Using the residual gauge transformation xi^(a)=iC^(a)exp(ik_(c)chi^(c))\xi^{a}=i C^{a} \exp \left(i k_{c} \chi^{c}\right), find an expression for the amplitude A_(ab)^(')A_{a b}^{\prime} of the gravitational wave in the chi^('a)\chi^{\prime a} coordinates.
d) Choosing k_(0)=k_(3)=1k_{0}=k_{3}=1, compute the values of the constants C^(a)C^{a} for which A_(0a)^(')=A_(3a)^(')=0A_{0 a}^{\prime}=A_{3 a}^{\prime}=0 and A_(a)^('a)=0A_{a}^{\prime a}=0.
Hint: Note that the harmonic gauge condition in itself puts some constraints on the form of A_(ab)A_{a b}.
Problem 2.133 Consider gravitational waves described by perturbations h_(mu nu)h_{\mu \nu} of the metric g_(mu nu)g_{\mu \nu} such that g_(mu nu)=eta_(mu nu)+h_(mu nu)g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}, where eta_(mu nu)\eta_{\mu \nu} is the Minkowski metric and |h_(mu nu)|≪1\left|h_{\mu \nu}\right| \ll 1.
a) In the transverse traceless gauge (TT gauge), the conditions h_(0i)=0h_{0 i}=0 and eta^(mu nu)h_(mu nu)=0\eta^{\mu \nu} h_{\mu \nu}=0 hold, which means that the Lorenz gauge condition reduces to del_(nu)h^(nu)_(mu)=0\partial_{\nu} h^{\nu}{ }_{\mu}=0. Using the TT gauge conditions, show that the number of independent components of h_(mu nu)h_{\mu \nu} is two.
b) Consider two particles at rest at (x,y,z)=(0,0,0)(x, y, z)=(0,0,0) and (x,y,z)=(L,0,0)(x, y, z)=(L, 0,0), respectively. A plus-polarized gravitational wave h_(+)h_{+}of frequency ff and amplitude h_(0)≪1h_{0} \ll 1 passes by, propagating in the zz-direction, such that
Show that the distance dd measured along the xx-axis between the two particles (i.e., the spatial separation along the equal tt hypersurface), as the wave passes, is given by
{:(2.135)d=[1-(1)/(2)h_(0)sin(2pi ft)]L.:}\begin{equation*}
d=\left[1-\frac{1}{2} h_{0} \sin (2 \pi f t)\right] L . \tag{2.135}
\end{equation*}
Problem 2.134 Assuming a source described by the energy-momentum tensor T^(mu nu)T^{\mu \nu}, the linearized Einstein equations are given by
and the solutions in terms of Green's functions can be written as bar(h)_(mu nu)(t,x)=4int(T_(mu nu)(t-|x-x^(')|,x^(')))/(|x-x^(')|)d^(3)x^(')∼(4)/(r)intT_(mu nu)(t-r,x^('))d^(3)x^(')\bar{h}_{\mu \nu}(t, \mathbf{x})=4 \int \frac{T_{\mu \nu}\left(t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right|, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} d^{3} x^{\prime} \sim \frac{4}{r} \int T_{\mu \nu}\left(t-r, \mathbf{x}^{\prime}\right) d^{3} x^{\prime},
where r-=|x|r \equiv|\mathbf{x}| is far away from the source. Using the conservation law for the energymomentum tensor, i.e., grad_(v)T^(mu nu)=0\nabla_{v} T^{\mu \nu}=0, show that the spatial components are
where rho=T^(00)\rho=T^{00} is the mass-energy density of the source.
Problem 2.135 a) For a neutron-star binary (with total mass M≃2.8M_(o.)M \simeq 2.8 M_{\odot} ) at a distance of 5 kpc with orbital period P=1hP=1 \mathrm{~h}, estimate the amplitude hh of the gravitational waves.
b) Again, for the same system, but now with P=0.02sP=0.02 \mathrm{~s} (giving f_(GW)=2f_("orbit ")=f_{\mathrm{GW}}=2 f_{\text {orbit }}= 100 Hz , which lies in the sensitive band of the gravitational wave observatory LIGO), estimate hh at a distance of 15 Mpc , which is approximately the distance to the Virgo cluster of galaxies.
c) Estimate the orbital separation RR when P=0.02sP=0.02 \mathrm{~s}.
d) For a neutron star at 1 kpc with a nonspherical deformation of mass delta M=\delta M=10^(-6)M_(o.)10^{-6} M_{\odot}, a spin frequency of 50 Hz , and a stellar radius of 10 km , determine the gravitational wave amplitude hh at Earth.
Problem 2.136 The first direct observation of gravitational waves was performed on September 14, 2015. This was presented by the LIGO and Virgo Collaborations on February 11, 2016, and was awarded the Nobel Prize in Physics in 2017. The original signal was named GW150914 and it was also the first observation of a binary black hole merger.
a) GW150914 had a maximal amplitude of h≃10^(-21)h \simeq 10^{-21} at a frequency of f≃f \simeq 200 Hz . Compute the corresponding energy flux at the Earth. The binary source of GW150914 is situated at an estimated distance of about 400 Mpc .
b) Estimate the energy flux in electromagnetic waves that is received at Earth from a full moon. Compare this energy flux to the gravitational wave energy flux of GW150914.
is equal to 1 in geometric units, i.e., c=1c=1 and G=1G=1.
2.12 Cosmology and Friedmann-Lemaître-Robertson-Walker Metric
Problem 2.137 Consider the linearly expanding spacetime with metric ds^(2)=d s^{2}=dt^(2)-H^(2)t^(2)dx^(2)d t^{2}-H^{2} t^{2} d x^{2}. You start at t=t_(0),x=0t=t_{0}, x=0 and want to arrive at x=Lx=L at t=t_(1)t=t_{1} without accelerating at any time. Find an expression for your position x(t)x(t) as a function of the global time tt. What is the largest LL you can arrive at in finite global time?
Problem 2.138 In a 2-dimensional mini-universe the metric element is given by
{:(2.140)ds^(2)=c^(2)dt^(2)-S(t)^(2)dchi^(2):}\begin{equation*}
d s^{2}=c^{2} d t^{2}-S(t)^{2} d \chi^{2} \tag{2.140}
\end{equation*}
where S(t)S(t) is some positive function of the time tt, constant in the "space" variable chi\chi. Explain the cosmological redshift and derive an expression for it by studying the emission and detection of light signals by comoving observers at two different locations chi_(0)\chi_{0} and chi_(1)\chi_{1}.
Problem 2.139 Consider the two-dimensional de Sitter universe with the metric ( c=1c=1 )
{:(2.141)ds^(2)=dt^(2)-e^(2t//R)dx^(2):}\begin{equation*}
d s^{2}=d t^{2}-e^{2 t / R} d x^{2} \tag{2.141}
\end{equation*}
where R > 0R>0 is a constant. Find an expression for the cosmological redshift between comoving observers at x_(0) > 0x_{0}>0 and x_(1) > x_(0)x_{1}>x_{0}.
Problem 2.140 Consider the two-dimensional de Sitter universe as defined in Problem 2.139.
a) Compute all nonzero Christoffel symbols for this metric.
b) Find the explicit form of the wave equation g_(mu nu)grad^(mu)grad^(nu)Phi=0g_{\mu \nu} \nabla^{\mu} \nabla^{\nu} \Phi=0, where Phi\Phi is a scalar field, in this universe.
Problem 2.141 Consider the Robertson-Walker spacetime described by the metric
{:(2.142)ds^(2)=dt^(2)-e^(2t//t_(H))[dr^(2)+r^(2)(dtheta^(2)+sin^(2)theta dvarphi^(2))]",":}\begin{equation*}
d s^{2}=d t^{2}-e^{2 t / t_{H}}\left[d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \varphi^{2}\right)\right], \tag{2.142}
\end{equation*}
with the coordinates (x^(mu))=(t,r,theta,varphi)\left(x^{\mu}\right)=(t, r, \theta, \varphi), where tt is the universal time and t_(H)~~14Gyrt_{H} \approx 14 \mathrm{Gyr} is the Hubble time.
a) Compute the path r(t)r(t) of a light pulse emitted at time t=t_(0)t=t_{0} at the origin r=0r=0. You may assume that the light ray moves along the line theta=pi//2\theta=\pi / 2 and varphi=0\varphi=0 so that dr//dt > 0d r / d t>0.
b) Compute the proper distance between the origin r=0r=0 and a point r > 0r>0 on the line theta=pi//2\theta=\pi / 2 and varphi=0\varphi=0 at fixed universal time tt.
c) Compute the spectral shift of the light pulse in a) during the time between emission at the origin and arrival at a point r > 0r>0.
Hint: The spectral shift is defined as z=lambda_("rec ")//lambda_(em)-1z=\lambda_{\text {rec }} / \lambda_{\mathrm{em}}-1, where lambda_(em)\lambda_{\mathrm{em}} is the wavelength of the light at emission and lambda_("rec ")\lambda_{\text {rec }} its wavelength when it arrives.
where a^(˙)(t)=da//dt\dot{a}(t)=d a / d t and v_(p)v_{p} and d_(p)d_{p} are the proper velocity and the proper distance, respectively, from the Robertson-Walker metric
{:(2.144)ds^(2)=dt^(2)-a(t)^(2)G_(ij)dx^(i)dx^(j):}\begin{equation*}
d s^{2}=d t^{2}-a(t)^{2} G_{i j} d x^{i} d x^{j} \tag{2.144}
\end{equation*}
b) What are the physical consequences of Hubble's law?
Problem 2.143 a) The first Friedmann equation can be written as
which can also be written as 1-Omega=-k//a^(˙)^(2)1-\Omega=-k / \dot{a}^{2}, where Omega=rho//rho_(c)\Omega=\rho / \rho_{c}. Assume now that kk is small compared to the energy density rho\rho, which mainly consists of the cosmological constant rho_(Lambda)\rho_{\Lambda}, thus leading to an inflationary universe. Show that the longer this scenario is assumed to last, the closer Omega\Omega gets to one.
b) Describe the flatness problem in words and how it is solved by inflation.
Problem 2.144 Our present universe can roughly be described by a spatially flat Friedmann-Lemaître-Robertson-Walker spacetime with Omega_(Lambda)=0.7,Omega_(m)=0.3\Omega_{\Lambda}=0.7, \Omega_{m}=0.3, Omega_(r)≃10^(-4)\Omega_{r} \simeq 10^{-4}, being the density parameters of the cosmological constant (dark energy), matter, and radiation, respectively.
a) How much smaller was the scale factor when the energy density of the dark energy was equal to the energy density of matter?
b) At what redshift zz did the matter-radiation equality (equal amounts of radiation and matter energy density) occur?
Problem 2.145 Our current universe is roughly described by a dark energy component Omega_(Lambda)=0.7\Omega_{\Lambda}=0.7 and a matter component Omega_(m)=0.3\Omega_{m}=0.3. Determine an integral expression for the future behavior of the scale factor a(t)a(t). You may assume that the scale factor today is a_(0)=1a_{0}=1 and that the current Hubble parameter is H_(0)H_{0}. Plot the result of your integral for 0 <= a(t) <= 1000 \leq a(t) \leq 100. Compare your result to the analytic result a(t)=a(t)=exp(H_(0)(t-t_(0)))\exp \left(H_{0}\left(t-t_{0}\right)\right) for Omega_(Lambda)=1\Omega_{\Lambda}=1 and Omega_(m)=0\Omega_{m}=0 and determine the age of the universe t_(0)t_{0} if a(0)=0a(0)=0.
Hint: You may use numerical integration.
Problem 2.146 A toy model 1+1-dimensional circular Robertson-Walker universe has the line element
{:(2.146)ds^(2)=dt^(2)-a(t)^(2)dvarphi^(2):}\begin{equation*}
d s^{2}=d t^{2}-a(t)^{2} d \varphi^{2} \tag{2.146}
\end{equation*}
where varphi\varphi and varphi+2n pi(n inN)\varphi+2 n \pi(n \in \mathbb{N}) correspond to the same spatial point. An object is thrown from varphi=varphi_(0)\varphi=\varphi_{0} at time t_(0)t_{0} with a velocity vv relative to a comoving observer.
a) Find a condition that must be satisfied in order for the object to complete a full lap around the universe to reach the comoving observer again from the other direction.
b) What is the relative velocity between the object and a comoving observer at an arbitrary time tt ?
You may assume that the scale factor a(t)a(t) has a known dependence on tt.
Problem 2.147 Consider a flat universe containing only matter and radiation components such that the radiation density today corresponds to Omega_(rad)=x≪1\Omega_{\mathrm{rad}}=x \ll 1. Find an expression for the time that has passed since matter and radiation had equal energy densities.
Problem 2.148 Consider a Friedmann-Lemaittre-Robertson-Walker universe with curvature parameter kappa!=0\kappa \neq 0. Determine the condition on the equation-of-state parameter w=p//rhow=p / \rho such that the curvature parameter |Omega_(K)|=|-1//(Ha)^(2)|\left|\Omega_{K}\right|=\left|-1 /(H a)^{2}\right| decreases with cosmological time tt for an expanding universe ( a^(˙) > 0\dot{a}>0 ).
Problem 2.149 For a flat single-component Friedmann-Lemaître-Robertson-
Walker universe with an arbitrary, but fixed, equation-of-state parameter w=p//rhow=p / \rho :
a) Compute the scale factor a(t)a(t) as a function of cosmological time tt.
b) Compute the Hubble parameter H(t)H(t) as a function of tt.
Hint: You may assume that, for some cosmological time t=t_(0)t=t_{0}, we normalize our parameters such that a_(0)=a(t_(0))=1a_{0}=a\left(t_{0}\right)=1 and H_(0)=H(t_(0))H_{0}=H\left(t_{0}\right) are known. Your answers should be given in terms of omega,t_(0)\omega, t_{0}, and H_(0)H_{0}.
Problem 2.150 A scalar field phi\phi with potential energy density V(phi)V(\phi) has a Lagrangian density given by
a) Derive the equation of motion for the scalar field phi\phi.
b) Assuming that the N+1N+1-dimensional spacetime has a metric given by
{:(2.148)ds^(2)=g_(mu nu)dx^(mu)dx^(nu)=dt^(2)-a(t)^(2)G_(ij)dx^(i)dx^(j):}\begin{equation*}
d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu}=d t^{2}-a(t)^{2} G_{i j} d x^{i} d x^{j} \tag{2.148}
\end{equation*}
where G_(ij)G_{i j} are the metric components on an NN-dimensional Riemannian manifold, and that the scalar field phi\phi only depends on the time coordinate tt, show that the scalar field is an ideal fluid and find the (time-dependent) equation-of-state parameter w=p//rho_(0)w=p / \rho_{0}.
^(1){ }^{1} Note the abuse of notation - the symbol x^(2)x^{2} denotes both the 'length' of the vector xx and the second spatial contravariant component of the vector xx. Unfortunately, this type of abuse of notation is difficult to avoid in relativity theory, since the notation would otherwise be too cumbersome.
